NONLINEAR FINITE ELEMENT STUDY OF DETERIORATED …
Transcript of NONLINEAR FINITE ELEMENT STUDY OF DETERIORATED …
NONLINEAR FINITE ELEMENT STUDY OF DETERIORATED RIGID
SEWERS INCLUDING THE INFLUENCE OF EROSION VOIDS
by
ZHENG TAN
A thesis submitted to the Department of Civil Engineering
in conformity with the requirements for
the degree of Master of Science (Engineering)
Queen’s University
Kingston, Ontario, Canada
September, 2007
Copyright © Zheng Tan, 2007
ABSTRACT
The service life of rigid sewer pipes is often controlled by joint integrity. Leaking joints
can cause ingress of water and develop voids where surrounding soil has eroded. The
influence of soil voids on the stability of buried rigid pipes is investigated, considering
the effects of void size, void location and void shape.
A series of simplified void geometries are defined, and their influence on bending
moments in the rigid sewer is studied through finite element analysis. Elastic analysis
indicates that the bending moments from expanding voids at the springline will increase
slowly, accelerating once the void spans a 45 degree arc, approximately doubling at 90
degrees, and tripling if the loosened backfill is modeled for shear failure. This
preliminary study suggests that the growth of erosion voids should be stopped before they
reach 45 degrees, but validation through physical testing is necessary.
Elastic-plastic finite element analysis is used to calculate the deformation of rigid
fractured pipe with different thicknesses, considering both bonded and full-slip interface
conditions. The analysis confirms that bonded idealized flexible pipe theory is very
effective for calculation of increases in horizontal diameter of the fractured pipe.
Furthermore, decreases in vertical diameter can be simply related to increase in
horizontal diameter using (1-2t/OD) obtained from fractured pipe kinematics.
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Both elastic and elastic-plastic finite element analyses used to study the deformations of
fractured rigid pipe reveal that contact angle appears to be the dominant factor affecting
fractured pipe deformations. Deformation of the damaged rigid pipe increases
dramatically with void growth and accelerates when erosion void contacts with the outer
surface of the pipe over an arc greater than 45 degrees. Computational analyses examine
the behavior of centrifuge model tests which examine soil load transfer to flexible sewer
liners after fracture and erosion voids form nearby. The magnitude of deformation
changes for finite element models is found to be comparable to observations when voids
are formed at springline. However the development patterns are dramatically different as
voids located under the invert, and it appears that the laboratory test featured physical
characteristics that are not modeled in the analysis.
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ACKNOWLEDGMENTS
First, I thank my supervisor, Prof. Ian Moore, for many insightful conversations during
the development of the ideas in this thesis, helpful comments on the text, technical
guidance throughout the course of this research, financial support, stimulating
suggestions, and encouragement.
I would also like to thank all the professors, staff, and colleagues at the department of
Civil Engineering, Queen’s University, for their support and friendship. Specially thanks
to the people who have taught me in Geotechnical Engineering: Dr. R. Brachman and Dr.
Andrew Take at Queen’s University; Dr. R. J. Bathurst and Dr. G. Akhras at Royal
Military College of Canada.
I wish to thank my friends, J. Ye, A.G. Chehab and H. Xia at Queen’s University for
helping me get through the difficult times, and for all the support and caring they
provided.
Finally, and most importantly, I wish to thank my parents, J.L. Tan and S.R. Cai. They
raised me, supported me, taught me, and loved me. To them I dedicate this thesis.
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TABLE OF CONTENTS ABSTRACT...................................................................................................................................... iiACKNOWLEDGMENTS ............................................................................................................... ivCHAPTER 1 INRODUCTION............................................................................................................................... 1 1.1 GENERAL.................................................................................................................................. 1 1.2 THEORY & DESIGN FOR PIPES IN UNIFORM SOIL.......................................................... 2 1.3 REPONSE OF FRACTURED SEWERS ................................................................................... 6
1.3.1 Rigid Pipe Fracture Configuration & Deformation Mechanisms.................................... 6 1.3.2 Kinematic Response of Fractured Rigid Pipe under Earth Loads ................................... 6
1.4 USE OF LINERS FOR SEWER REPAIR AND LINER RESPONSE TO EARTH LOADS ............................................................................................................................... 10
1.5 OBJECTIVES........................................................................................................................... 12 1.6 OUTLINE OF THESIS............................................................................................................. 13 CHAPTER 2 EFFECT OF BACKFILL EROSION ON MOMENTS IN BURIED RIGID PIPES..................... 24 2.1 INTRODUCTION .................................................................................................................... 24 2.2 THE RIGID PIPE ..................................................................................................................... 25 2.3 NUMERICAL MODELING – CONCRETE PIPE ALONE.................................................... 27 2.4 NUMERICAL MODELING - ELASTIC CALCULATIONS FOR VOIDS AT
SPRINGLINES................................................................................................................... 28 2.5 NUMERICAL MODELING - ELASTIC-PLASTIC CALCULATIONS FOR VOIDS
AT SPRINGLINES............................................................................................................. 31 2.6 NUMERICAL MODELING - ELASTIC-PLASTIC CALCULATIONS FOR VOIDS
AT INVERT........................................................................................................................ 32 2.7 DISCUSSION AND CONCLUSIONS .................................................................................... 33 CHAPTER 3 NUMERICAL STUDY OF EFFECT OF THICKNESS ON THE DEFORMATION OF
FRACTURED PIPE: RESPONSE TO EARTH LOADS................................................... 56 3.1 INTRODUCTION .................................................................................................................... 56 3.2 DESCRIPTION OF THE FINITE ELEMENT MODELING .................................................. 57 3.3 MODELING PIPES WITH DIFFERENT THICKNESS......................................................... 58 3.4 SOIL MODELING AND LOADING SEQUENCE................................................................. 59 3.5 RESULTS ................................................................................................................................. 61 3.6 IMPLICATION FOR LINER DESIGN ................................................................................... 63 3.7 SUMMARY AND CONCLUSION ......................................................................................... 65 CHAPTER 4 NUMERICAL STUDY OF DEFORMATIONS IN FRACTURED SEWERS: RESPONSE
TO EARTH LOADS AND EROSION VOIDS ................................................................. 78 4.1 INTRODUCTION .................................................................................................................... 78
4.2.1 Introduction to The Finite Element Modeling............................................................... 79 4.2.2 Modeling of Soil and Erosion Voids ............................................................................. 80 4.2.3 Analysis Sequence......................................................................................................... 81 4.2.4 Results ........................................................................................................................... 82 4.2.5 Discussion ..................................................................................................................... 83 4.2.6 Interpretation of Pipe Deflections to Infer ‘Damaged’ Soil Condition.......................... 84
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4.3 FINITE ELEMENT ANALYSIS OF CENTRIFUGE TESTS EXAMINING SOIL LOAD TRANSFER TO FLEXIBLE SEWER LINER....................................................... 86
4.3.1 Description of The Centrifuge Modeling ...................................................................... 86 4.3.2 Modeling of Host-Pipe Liner System Information........................................................ 87 4.3.3 Numerical Modeling Sequence ..................................................................................... 90 4.3.4 Results and Discussion.................................................................................................. 91
4.4 SUMMARY AND CONCLUSION ......................................................................................... 94 CHAPTER 5 DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS.............................................. 110 5.1 DISCUSSION AND CONCLUSIONS .................................................................................. 110 5.2 RECOMMENDATIONS........................................................................................................ 112 APPENDIX A FINITE ELEMENT ANALYSIS OF BURIED RC PIPE TEST – ELASTIC ANALYSIS
(VOIDS AT SPRINGLINE) ............................................................................................. 115 A.1 DESCRIPTION OF THE FINITE ELEMENT MODELING ............................................... 115 A.2 MODELING OF PIPE AND SOIL INFORMATION........................................................... 117 A.3 MODELING OF EROSION VOIDS INFORMATION........................................................ 117 A.4 TEST SEQUENCE ................................................................................................................ 118 A.5 RESULTS .............................................................................................................................. 118 A.5 RESULTS ANALYSIS.......................................................................................................... 120 APPENDIX B FINITE ELEMENT ANALYSIS OF BURIED PIPE TEST – ELASTO-PLASTIC
ANALYSIS (VOIDS AT SPRINGLINE) ........................................................................ 140 B.1 DESCRIPTION OF THE FINITE ELEMENT MODELING................................................ 140 B.2 MODELING OF SOIL INFORMATION.............................................................................. 140 B.3 MODELING OF EROSION VOIDS INFORMATION ........................................................ 140 B.4 RESULTS .............................................................................................................................. 141 B.5 RESULTS ANALYSIS.......................................................................................................... 143 APPENDIX C FINITE ELEMENT ANALYSIS OF BURIED RC PIPE TEST – ELASTO-PLASTIC
ANALYSIS (VOIDS AT INVERT) ................................................................................. 162 C.1 DESCRIPTION OF THE FINITE ELEMENT MODELING................................................ 162 C.2 MODELING OF SOIL INFORMATION.............................................................................. 163 C.3 RESULTS .............................................................................................................................. 163 C.4 RESULTS ANALYSIS.......................................................................................................... 164 APPENDIX D SYMBOLS AND ACRONYMS .................................................................................................. 171
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LIST OF TABLES
Table 2.1 Calculation Results of the Maximum Bending Moment, Maximum Stress and
Maximum Deflection of the RC Pipe with Parallel Plate Load 23KN/m..................................... 37 Table 4.1 List of Different Finite Element Models (MB1 & MB2)................................................... 97 Table 4.2 List of Different Finite Element Models (HS1 & HI1)...................................................... 97 Table A.1 List of Different Finite Element Models......................................................................... 130 Table A.2 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of
MC1................................................................................................................................. 130 Table A.3 Results of Increasing Ratio Based on Table A.2............................................................. 130 Table A.4 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
MC1................................................................................................................................. 131 Table A.5 Results of Increasing Ratio Based on Table A.4............................................................. 131 Table A.6 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline
of MC1 ............................................................................................................................ 131 Table A.7 Results of Increasing Ratio Based on Table A.6............................................................. 132 Table A.8 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of
MC2................................................................................................................................. 132 Table A.9 Results of Increasing Ratio Based on Table A.8............................................................. 132 Table A.10 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
MC2................................................................................................................................. 133 Table A.11 Results of Increasing Ratio Based on Table A.10......................................................... 133 Table A.12 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline
of MC2 ............................................................................................................................ 133 Table A.13 Results of Increasing Ratio Based on Table A.12......................................................... 134 Table A.14 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of
MC3................................................................................................................................. 134 Table A.15 Results of Increasing Ratio Based on Table A.14......................................................... 134 Table A.16 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
MC3................................................................................................................................. 135 Table A.17 Results of Increasing Ratio Based on Table A.16......................................................... 135 Table A.18 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline
of MC3 ............................................................................................................................ 135 Table A.19 Results of Increasing Ratio Based on Table A.18......................................................... 136 Table A.20 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (29.6°) – Bonded Interface ..................................... 136 Table A.21 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (29.6°) – Full-Slip Interface.................................... 137 Table A.22 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (59.7°) – Bonded Interface ..................................... 137 Table A.23 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (59.7°) – Full-Slip Interface.................................... 138 Table A.24 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (90.8°) – Bonded Interface ..................................... 138
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Table A.25 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing Areas @ Contact Angle (90.8°) – Full-Slip Interface.................................... 139
Table B.1 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of MD1 ................................................................................................................................ 153
Table B.2 Results of Increasing Ratio Based on Table B.1 ............................................................. 153 Table B.3 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
MD1 ................................................................................................................................ 153 Table B.4 Results of Increasing Ratio Based on Table B.3 ............................................................. 154 Table B.5 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Spring line
of MD1 ............................................................................................................................ 154 Table B.6 Results of Increasing Ratio Based on Table B.5 ............................................................. 154 Table B.7 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of
MD2 ................................................................................................................................ 155 Table B.8 Results of Increasing Ratio Based on Table B.7 ............................................................. 155 Table B.9 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
MD2 ................................................................................................................................ 155 Table B.10 Results of Increasing Ratio Based on Table B.9 ........................................................... 156 Table B.11 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Spring
line of MD2 ..................................................................................................................... 156 Table B.12 Results of Increasing Ratio Based on Table B.11 ......................................................... 156 Table B.13 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of
MD3 ................................................................................................................................ 157 Table B.14 Results of Increasing Ratio Based on Table B.12 ......................................................... 157 Table B.15 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
MD2 ................................................................................................................................ 157 Table B.16 Results of Increasing Ratio Based on Table B.15 ......................................................... 158 Table B.17 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Spring
line of MD3 ..................................................................................................................... 158 Table B.18 Results of Increasing Ratio Based on Table B.16 ......................................................... 158 Table B.19 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (29.6°) – Bonded Interface ..................................... 159 Table B.20 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (29.6°) – Full-Slip Interface.................................... 159 Table B.21 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (59.7°) – Bonded Interface ..................................... 160 Table B.22 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (59.7°) – Full-Slip Interface.................................... 160 Table B.23 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (90.8°) – Bonded Interface ..................................... 161 Table B.24 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with
Increasing Areas @ Contact Angle (90.8°) – Full-Slip Interface.................................... 161 Table C.1 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of
ME1................................................................................................................................. 169 Table C.2 Results of Increasing Ratio Based on Table C.1 .............................................................. 169 Table C.3 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
ME1................................................................................................................................. 169
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Table C.4 Results of Increasing Ratio Based on Table C.3 .............................................................. 170 Table C.5 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline
of ME1............................................................................................................................. 170 Table C.6 Results of Increasing Ratio Based on Table C.5 .............................................................. 170
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LIST OF FIGURES Figure 1.1 Definition of Terms for the Pipe......................................................................................... 18 Figure 1.2 External Loads Acting Directly on the Circular Pipes (Moore 2001) ................................ 18 Figure 1.3 Thrust and Bending Moment at Crown of Pipe.................................................................. 19 Figure 1.4 Rigid Pipe with Overloading Fractures (ATV-M 143-2, with Permission) ....................... 19Figure 1.5 Kinematics of Fractured Rigid Pipe (after Law, 2004) ...................................................... 20 Figure 1.6 Pipe Deformation Measurements and Theoretical Deformations....................................... 21 Figure 1.7 Comparison of Finite Element Analysis and Laboratory Measurements (after
Law, 2004) ........................................................................................................................ 21 Figure 1.8 Possible Three Dimensional Void Geometry Due to Water Ingress at Joint...................... 22Figure 1.9 Plane Strain Idealization of Void Geometry for Preliminary Analysis .............................. 22Figure 1.10 Possible Three Dimensional Void Geometry Due to Ingress at Joint and
through Longitudinal Fractures ......................................................................................... 23 Figure 1.11 Plane Strain Idealization of Void Geometry Due to Ingress at Joint and
through Longitudinal Fractures ......................................................................................... 23 Figure 2.1 Vertical Earth Loads at the Pipe Crown and Invert (2 Point Loading and 3 Point
Loading) ............................................................................................................................ 38 Figure 2.2 Finite Element Analysis of Rigid Pipe Alone (Tension Negative)..................................... 39 Figure 2.3 Finite Element Models of Buried RC Pipe with Voids at Springline................................. 40 Figure 2.4 Void Geometry for Void Set A, B & C .............................................................................. 41 Figure 2.5 Stress xxσ at Crown of the Rigid Pipe – Void Set A – Elastic (Compress
Positive)............................................................................................................................. 42 Figure 2.6 Stress xxσ at Invert of the Rigid Pipe – Void Set A - Elastic (Compress
Positive)............................................................................................................................. 43 Figure 2.7 Stress yyσ at Springline of the Rigid Pipe – Void Set A – Elastic (Compress
Positive)............................................................................................................................. 44 Figure 2.8 Elastic Analysis of Increases in Extreme Fiber Stresses at the Crown as a
Function of Void Geometry and Interface Condition........................................................ 45 Figure 2.9 Stress Comparison (@ Crown and Springline) between Elastic Analysis and
Elastic- Plastic Analysis for Bonded Interface (E – Elastic, EP – Elastic-Plastic) ............................................................................................................................... 46
Figure 2.10 Stress Comparison (@ Crown and Springline) between Elastic Analysis and Elastic- Plastic Analysis for Smooth Interface (E – Elastic, EP – Elastic-Plastic) ............................................................................................................................... 47
Figure 2.11 Stress xxσ at Crown of the Rigid Pipe – Void Set A – Elastic-Plastic (Compress Positive) .......................................................................................................... 48
Figure 2.12 Stress xxσ at Invert of the Rigid Pipe – Void Set A - Elastic-Plastic (Compress Positive)............................................................................................................................. 49
Figure 2.13 Stress yyσ at Springline of the Rigid Pipe - Void Set A - Elastic-Plastic (Compress Positive) .......................................................................................................... 50
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Figure 2.14 Elastic-plastic Analysis of Increases in Extreme Fiber Stresses at the Crown as a Function of Void Geometry and Interface Condition..................................................... 51
Figure 2.15 Mesh Defining Voids under Invert (Void Set D) ............................................................. 52 Figure 2.16 Stress xxσ at Crown of the Rigid Pipe – Void Set D – Elastic-Plastic
(Compress Positive) .......................................................................................................... 53 Figure 2.17 Stress xxσ at Invert of the Rigid Pipe – Void Set D – Elastic-Plastic
(Compress Positive) .......................................................................................................... 54 Figure 2.18 Stress yyσ at Springline of the Rigid Pipe – Void Set D – Elastic-Plastic
(Compress Positive) .......................................................................................................... 55 Figure 3.1 Idealized Deformation of Damaged Pipes – before Disturbance (after Law and
Moore, 2004)..................................................................................................................... 67 Figure 3.2 Idealized Deformation of Damaged Pipes – after Disturbance (after Law and
Moore, 2004)..................................................................................................................... 67 Figure 3.3 Finite Element Model of Buried Reinforced Concrete Pipe (MA1 & MA2) ..................... 68Figure 3.4 Finite Element Model Details of Buried Reinforced Concrete Pipe (MA1 &
MA2) ................................................................................................................................. 69 Figure 3.5 Finite Element Model Details of Buried Reinforced Concrete Pipe (MA1 &
MA2) ................................................................................................................................. 70 Figure 3.6 Detail “A”........................................................................................................................... 71 Figure 3.7 Detail “B” ........................................................................................................................... 72 Figure 3.8 Detail “C” ( eqtt = ) ............................................................................................................ 73
Figure 3.9 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Bonded Interface.............................................................................................. 74
Figure 3.10 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Full-Slip Interface ............................................................................................ 74
Figure 3.11 Horizontal and Vertical Deformation of Rigid Pipe Comparison between Full-Slip Interface and Bonded Interface .................................................................................. 75
Figure 3.12 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Bonded Interface.............................................................................................. 75
Figure 3.13 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Full-Slip Interface ............................................................................................ 76
Figure 3.14 Results Compassion between Idealized Flexible Pipe Theory and Numerical Modeling Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Full-Slip Interface .......................................................................................................... 76
Figure 3.15 Horizontal and Vertical Deformation of Rigid Pipe Comparison between Full-Slip Interface and Bonded Interface .................................................................................. 77
Figure 4.1 Finite Element Model of Buried Reinforced Concrete Pipe with Voids at Springline (MB) ................................................................................................................ 98
Figure 4.2 Finite Element Mesh Shows Details of Buried Reinforced Concrete Pipe with Voids at Springline (MB); Contact Angle Illustrated for void MBV4.............................. 99
Figure 4.3 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Angular Contact between Pipe and Voids at Springline (MB1) ................................................... 100
Figure 4.4 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Angular Contact between Pipe and Voids at Springline (MB2) ................................................... 101
Figure 4.5 Estimation of Changes in Horizontal Diameter – Option 2.............................................. 102
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Figure 4.6 Plan Layout of the Centrifuge Package, Spasojevic et al. (2007)..................................... 103 Figure 4.7 Cross-section of the Model, Spasojevic et al. (2007) ....................................................... 103 Figure 4.8 The Numerical Model of the Pipe-Liner-Soil System...................................................... 104 Figure 4.9 Host Pipe-Liner-Soil Finite Element Mesh ...................................................................... 105 Figure 4.10 Elements in Parts of the Void Removed, Namely a Zone of Elements Closest
to Pipe Springline ............................................................................................................ 106 Figure 4.11 Analysis of Diameter Changes for Models HS1 and HI1............................................... 107 Figure 4.12 Results Comparison between Lab Test and Numerical Method Analysis
(Model HS1).................................................................................................................... 108 Figure 4.13 Results Comparison between Lab Test and Numerical Method Analysis
(Model HI1)..................................................................................................................... 109 Figure A.1 Stress xxσ of Soil – MC1 (No Void) ............................................................................ 122
Figure A.2 Stress xxσ of Soil – MC1 (Remove Void1) .................................................................. 123
Figure A.3 Stress xxσ of Soil – MC1 (Remove Void2) .................................................................. 124
Figure A.4 Stress xxσ of Soil – MC1 (Remove Void3) .................................................................. 125
Figure A.5 Stress yyσ of Soil – MC1 (No Void) ............................................................................ 126
Figure A.6 Stress yyσ of Soil – MC1 (Remove Void1) .................................................................. 127
Figure A.7 Stress yyσ of Soil – MC1 (Remove Void2) .................................................................. 128
Figure A.8 Stress yyσ of Soil – MC1 (Remove Void3) .................................................................. 129
Figure B.1 Stress xxσ of Soil – MD1 (No Void) ............................................................................ 145
Figure B.2 Stress xxσ of Soil – MD1 (Remove Void1).................................................................. 146
Figure B.3 Stress xxσ of Soil – MD1 (Remove Void2).................................................................. 147
Figure B.4 Stress xxσ of Soil – MD1 (Remove Void3).................................................................. 148
Figure B.5 Stress yyσ of Soil – MD1 (No Void) ............................................................................ 149
Figure B.6 Stress yyσ of Soil – MD1 (Remove Void1) .................................................................. 150
Figure B.7 Stress yyσ of Soil – MD1 (Remove Void2) .................................................................. 151
Figure B.8 Stress yyσ of Soil – MD1 (Remove Void3) .................................................................. 152
Figure C.1 Stress xxσ of Soil – Bonded Condition (No Void) - Left .............................................. 165
Figure C.2 Stress xxσ of Soil – Bonded Condition (Remove Void1) - Right.................................. 165
Figure C.3 Stress xxσ of Soil – Bonded Condition Remove Void2) - Left ..................................... 166
Figure C.4 Stress xxσ of Soil – Bonded Condition Remove Void3) - Right ................................... 166
Figure C.5 Stress yyσ of Soil – Bonded Condition (No Void) - Left .............................................. 167
Figure C.6 Stress yyσ of Soil – Bonded Condition (Remove Void1) - Right.................................. 167
Figure C.7 Stress yyσ of Soil – Bonded Condition (Remove Void2) - Left .................................... 168
Figure C.8 Stress yyσ of Soil – Bonded Condition (Remove Void3) - Right.................................. 168
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CHAPTER 1
INRODUCTION
1.1 GENERAL
Buried pipe infrastructure is used to supply water, sewage, electricity, gas and oil to our
communities and resource industries. The pipes surrounded by soil are both loaded and
supported by the earth and pore water. Generally, the zone of backfill soil placed directly
adjacent to the pipe is specially selected to provide a stable supporting environment, and
often consists of granular material. The soil placed beneath the pipe is also specially
prepared. Figure 1.1 defines key locations around the pipe circumference. The crown,
springlines and invert are generally those positions where the pipe reaches critical limit
states, though shoulders and haunches can also be critical as a result of non-uniform
surface loading or narrow support at the invert (Moore, 2001).
Because of the wide variety of materials used in pipe manufacture, the stiffness, strength,
ductility and durability characteristics of pipes can vary significantly. The variations in
pipe wall geometry will lead to further complexity. A rigid pipe may be defined as one
that, under its maximum load, does not deform sufficiently to produce a significant
amount of passive restraint at the sides from the soil in which it is laid. With its high hoop
stiffness, a rigid pipe supports loads in the ground as a ring in bending, and in some ways
is analogous to a beam.
In the industrial market, the most common rigid pipes are clay, cast iron, unreinforced
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concrete and reinforced concrete pipes, etc. The development of concrete pipes dates
from soon after the introduction of Portland cement in 1845. Steel reinforcement for
concrete pipes was produced in Germany as early as the 1920s, but its use was not widely
accepted until after World War II. Today, concrete pipes are widely used internationally
for sewerage and drainage. The research in this thesis is mainly focused on reinforced
concrete pipes, which provide conservative calculations of deformation given that it is
assumed that the steel is exposed and corrodes rapidly so that circumferential moment is
fully released across cracks.
1.2 THEORY & DESIGN FOR PIPES IN UNIFORM SOIL
Various theoretical solutions have been published to estimate the response of a pipe of
radius r, modulus and section properties and when it is buried in a uniform
soil of modulus and Poisson’s ratio
pE pI pA
sE sν . Traditionally, soil-pipe interaction researchers
have solved for two idealized interface conditions: (a) perfect adhesion of the soil to the
pipe structure (the perfectly rough, no-slip, or bonded interface condition): and (b) zero
adhesion (the full slip or smooth interface condition). Since the actual pipe response in
the field is expected to lie somewhere between these two limits, intermediate response
may be evaluated using numerical solutions that account for the finite shear strength of
the pipe-soil interface (e.g. Katona 1978).
Hoeg (1968) developed a general pipe-soil interaction solution for the case where
vh Kσσ = (for arbitrary K), which can be used for understanding buried circular pipe.
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Moore (1988) developed related solutions for both the initially unstressed and prestressed
ground cases.
In general, thrust and moment can be estimated once interface pressure components 0σ ,
2σ and 2τ have been evaluated, as shown on Figure 1.2. Moore (2001) indicates that
thrusts at the crown, , and springline, , are given by crN spN
rrNcr )3/23/( 220 τσσ +−= (1.1)
rrN sp )3/23/( 220 τσσ ++= (1.2)
while bending moments at crown, , and the springline, , are given by crM spM
222 )6/3/( rM cr τσ += (1.3)
222 )6/3/( rM sp τσ +−= (1.4)
If a section is cut at crown of a pipe, the thrust and bending moment are shown on Figure
1.3.
For buried rigid pipe, C and F are small, where these two stiffness parameters are defined
by Hoeg (1968). For this stiffness limit, thrust at the spring line, , and bending
moment at the spring line, , are given by
spN
spM
)]43/()1(21)[1( ssvsp KKrN ννσ −−++−= Bonded interface (1.5)
)]65/()1(21)[1( ssvsp KKrN ννσ −−++−= Smooth interface (1.6)
)43/()1)(1(2ssvsp KrM ννσ −−−−= Bonded interface (1.7)
)65/(2)1)(1(2ssvsp KrM ννσ −−−−= Smooth interface (1.8)
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Rigid Pipe Design
Structural design of rigid pipe is primarily focused on preventing fractures in
unreinforced pipes, and limiting crack width at points of peak bending moment in steel
reinforced pipes (the limit for crack width is generally specified in the pipe design
standards). Moore (2001) discusses the two rigid pipe design procedures, denoted
“direct” and “indirect”. These essentially relate the vertical force that can act across the
vertical diameter of a rigid pipe during a standard two or three point bending test, , to
the vertical earth load supported by the buried pipe.
crW
vW
In indirect design, the vertical earth load is related to the factored vertical cracking
load using the empirical bedding factor BF:
vW
SFWcr /
BFWSFW vcr // = (1.9)
In direct design, springline moment associated with the limiting cracking can be
estimated from the vertical cracking load , which assumes the pipe sample responds
as an elastic ring.
crW
SFrWM crsp /)5.0( 1 −= −π (1.10)
Soil structure interaction analysis is used to estimate based on burial condition (in
particular, the lateral earth pressure ratio K). Rational (i.e. “direct”) design methods can
spM
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be developed for any standard bedding condition provided standard definitions of K (BF)
are established. More comprehensive earth pressure distributions and their associated
peak moment values have been proposed and implemented for RC pipe design, McGrath
and Kurdziel (1991).
When rigid pipes are correctly designed and installed, their deflection is negligible.
Deflection can be significant and is an important design consideration for non-rigid pipes;
and is largely controlled by the stiffness of the backfill, which is supporting the buried
structure.
For a flexible pipe, which is stiff in hoop compression (Hoeg’s stiffness parameters F
large and C negligible)
])23/[()1)(1)(1(8 ssssvv EKrD νννσ −+−−−=∆ (1.11)
For a rigid pipe (F and C negligible)
]/[)1)(1(2)]43(3/[)1)(1(2 24ppsvsppsvv IEKrIEKrD νσννσ −+−−−−−=∆ (1.12)
Based on the many advances in new pipe analysis and design, it would at first appear that
the challenges associated with the buried rigid pipe problem have been resolved.
However considering durability and mechanisms of failure over time, many rigid pipes
fail as a result of joint leakage, ingress of groundwater (a problem where water seeps into
the pipe at the joints, causing erosion voids around the pipe as nearby soil settles), and the
accompanying erosion of the soil envelope adjacent to the pipe. Further research is
needed to study the physical processes and implications of deterioration of the backfill
5
soil. Analyses are needed to provide guidance on the acceptable limits of soil erosion, to
explore the impact of soil voids on the bending moments in buried rigid pipes, and
laboratory tests to study deteriorated soil-pipe systems.
1.3 REPONSE OF FRACTURED SEWERS
1.3.1 Rigid Pipe Fracture Configuration & Deformation Mechanisms
As mentioned above, a pipe is considered “rigid” when its bending and hoop stiffness are
very large relative to the stiffness of the surrounding soil. Pipe fractures can result in a
loss of both hydraulic and structural capacity. In Europe, as a consequence of unequal
loading or variable soil conditions, Buco et. (2007a,b) studied the longitudinal bending of
buried gravity pipes and the circumferential pipe cracking or joint opening which results
in groundwater infiltration or leakage. To quantify the nature and frequency of damage
associated with this longitudinal pipe behavior, they undertook a statistical analysis of
CCTV inspections gathered in the Greater Lyon region of France. This concluded that the
major part of the observed defects might be related to the longitudinal behavior. They
then developed a statistical model to identify factors contributing to these longitudinal
bending failures. They also developed a 3D numerical model and an experimental
apparatus to analyze both rigid and flexible joints under different load conditions.
However, circumferential pipe cracking or joint opening for underground sewers is much
less common in North America due to different design procedures (the use of steel
6
reinforcement and heavier wall sections in most concrete pipes, for example). Here, a
common failure mechanism is associated with “overloading fractures”, where
longitudinal cracks form in the vicinity of the crown, invert, and springlines of the pipe,
where circumferential bending moments are highest. When a fracture develops, the
bending moment at that location is fully or partially released at that location and
redistributed around the rest of the pipe ring. If the original bending moment at the
fracture location is high, a second fracture will develop 180o away, which causes further
redistribution of moment. Eventually, the pipe breaks into approximately four equal
quadrants, as shown on Figure 1.4. The fracture pattern separating the pipe into four
identical 90o segments represents the most flexible configuration that can form, and that
is the one with least resistance to non-uniform earth pressures, Law and Moore (2004).
This fracture configuration represents the worst case scenario that leads to the greatest
fractured pipe deformations, and will be the primary focus of attention in the remainder
of this thesis.
1.3.2 Kinematic Response of Fractured Rigid Pipe under Earth Loads
Shown on Figures 1.5a and 1.5b, are the geometry of undeformed pipe and the geometry
of the lower right hand quadrant following rotation at the invert. Law and Moore (2004)
analyzed the kinematics of this system. First, they concluded that C, which is shown on
Figure 1.5b, is a function of the inner pipe radius R and the pipe thickness tpipe
22)( RRtC pipe ++= (1.13)
The angle between chord BS and the vertical when the pipe is undeformed is defined as η
7
⎟⎟⎠
⎞⎜⎜⎝
⎛
+= −
RtR
pipe
1tanη (1.14)
For the deformed pipe, the fracture at the invert opens at the inner surface. Changes in
diameter are given by
( )ηδη sin)sin(2 −+⋅⋅=∆ CDh (1.15)
( )ηδη cos)cos(2 −+⋅⋅=∆ CDv (1.16)
For a fractured pipe with negligible thickness, Equations 1.15 and 1.16 yield equal and
opposite pipe deformations.
Lee (2002) conducted a full scale laboratory test to examine the response of a partially
fractured sewer pipe. He studied the fractured pipe deformation mechanism using a
physical model of the rigid pipe with four fractures and hinges, and evaluated the
deviation of vertical from horizontal diameter change. Results show that the theoretical
solution matches the physical model deformations well, and the deviation between the
vertical and horizontal diameter changes of the fractured pipe is small.
Law and Moore (2004) studied the response of fractured pipes under earth loads to
establish the relationship between the fractured pipe deformations and the soil modulus,
the effect of stress history on its behavior, and to develop a design equation to estimate
fractured pipe deformations.
Their results from a full scale laboratory test (Figure 1.6) indicate that the changes in
vertical and horizontal pipe diameter of a rigid pipe with overloading fractures are similar
8
to a flexible pipe (Equation 1.11), in that deformation is controlled by the surrounding
soil. Bonded idealized flexible pipe theory using secant soil modulus was found to be
very effective in calculating the fractured pipe deformations due to changes in earth
loads.
Law and Moore (2004) used finite element analysis to study the kinematics of the
fractured pipe, Figure 1.7, and results show that the fracture angular expansion is
identical at all four fractures locations, matching the theoretical kinematics (equations
1.13 to 1.17) and laboratory observations well. Finite element analyses of the buried pipe
test revealed that the fractured pipe-soil interface behaves more like the full-slip
idealization than the bonded case. However, forced adhesion between the pipe
segments and the soil produced substantial restraints at the springlines, causing
unrealistic resistance to pipe deformations.
However, Law and Moore (2004) did not conduct a general parametric study to quantify
the effect of rigid pipe thickness on those deformations. The kinematics of fracture pipe
response, captured in equations 1.15 and 1.16, imply that the changes in vertical and
horizontal pipe diameter are not equal (unless the pipe is very thin), so finite element
analyses are needed to reveal the relationship between deformation of a fractured rigid
pipe and its wall thickness.
9
1.4 USE OF LINERS FOR SEWER REPAIR AND LINER RESPONSE TO EARTH
LOADS
As mentioned in Section 1.2, the growing need to repair and replace aging culvert and
sewer infrastructure has made it clear that durability is not simply a function of the long
term performance of the structural material. Rather, many rigid pipes (i.e. reinforced
concrete) fail as a result of joint leakage, ingress of groundwater, and the accompanying
erosion of the soil envelope adjacent to the pipe.
Once again, computer study and laboratory tests are needed here to quantify the impact of
soil voids on the deflections in buried fractured rigid pipes. When idealized soil voids are
defined, two dimensional finite element analyses are used to assess the impact of those
voids on deflections.
High deterioration rates of aging pipeline infrastructure are becoming increasingly
common, and use of trenchless technologies is growing to minimizing the cost, duration
and disruption with pipeline replacement and renovation. The cured-in-place pipe (CIPP)
process provides a cost-effective and rapid method for trenchless renovation of
deteriorated sewer pipes. Cured-in-place pipe liners were introduced in the early 1970’s
(Gumbel, 2003), and their response to external groundwater pressure has been well
studied. However, there is relatively little research about liner response to ground
loading.
10
Law and Moore (2004) conducted laboratory tests and numerical models to study the
static response of fractured sewers repaired by close-fitting non-bonded HDPE liners to
vertical loads applied on the soil surface. They also compared the results of the liner
encased within the old sewer with the same HDPE liner buried alone in the soil. Law and
Moore showed that the response of a lined host pipe was stiffer than that of the liner and
that significant local bending can occur in the liner due to the local contacts between the
host pipe and the liner at the crown and invert. They concluded that the separation of the
contact loads into a pair at crown and invert as a result of angular expansion of the
fracture has a negligible effect on the local bending in the liner, and the liner stiffness has
a negligible effect on the deformation induced by the earth loads on the damaged sewer
pipe.
In order fill the gap in knowledge regarding the effect of erosion voids in the soil around
a fractured rigid pipe, Spasojevic et al. (2007) describe 1/30th scale physical model tests
in dry sand responding under 30g of centrifugal acceleration. The model included a) a
deteriorating host pipe with two modes of deterioration, b) an instrumented close-fitting,
non-bonded, flexible liner, and c) voids in the soil. His test results showed that, even
under extreme loading conditions, the deformations of the liner are relatively small (2-3%
of diameter), though the soil voids were not modeled as being in direct contact with the
old sewer. The response to external loads of the centrifuge models of the deteriorated pipe
rehabilitated using a flexible liner was governed primarily by interaction between the
existing soil-host pipe structure and the soil voids. The results also show that void
geometry and position are dominating factors affecting the soil-pipe interaction.
11
In order to further prove the conclusion that soil voids dominate the soil-pipe interaction,
the laboratory tests of Spasojevic et al. (2007) need further study (e.g. using finite
element methods).
1.5 OBJECTIVES
The effect of deterioration of the backfill soil around buried sewer pipes needs to be
studied in detail, since it is that deterioration that leads to the load capacity of the pipe
being exceeded, pipe fracture, and all subsequent deformations. The specific objectives of
this study are:
1. To evaluate the load-carrying capacity of rigid pipe where voids have developed in
the backfill soil beside and under the pipe, quantifying bending moment changes with
void growth
2. To examine pipe deflections after fractures develop, explaining and quantifying the
effects of wall thickness on the pipe deformations, and establishing how deflections
increase as voids grow
3. To understand previous experimental work on deteriorated rigid pipe, and evaluate
the performance of new computer models for the deteriorated pipe system
12
4. To develop preliminary guidance regarding laboratory testing needed to establish the
performance of the new computer models
1.6 OUTLINE OF THESIS
This thesis reports on research initiated to study the physical processes and implications
of deterioration of the backfill soil on buried rigid pipes. The work employs computer
analyses, to provide preliminary guidance on the acceptable limits of soil erosion, and to
help inform work to be undertaken by a subsequent investigator employing physical
model studies. The finite element method was used to examine the problem under a range
of different conditions and scenarios.
The second chapter quantifies the impact of soil voids on the bending moments in buried
rigid pipes. In real life, possible three dimensional void geometry ingress at joint is
shown on Figure 1.8, which is very difficult to simulate in numerical models; since the
specifics of the three dimensional void geometries are not know, this first study of
circumferential behaviour employs a two dimensional idealization of the soil voids, as
shown on Figure 1.9. Two dimensional finite element analyses are then used to assess the
impact of those voids on bending moments. Voids at the springlines and the invert are
studied, to determine moment amplification with increases in the extent of the void. The
study indicates how soil erosion and void growth influence pipe failure. Both elastic and
elastic-plastic analyses are used and results are presented in this chapter in order to
understand the influence of shear failure in the backfill.
13
Rigid pipes manufactured in North America employ large wall thickness, varying from
2.5 to 12.5% of the pipe diameter. Chapter Three presents a finite element study aimed at
understanding how wall thickness affects the fractured pipe deformations under earth
loads.
In Chapter Four reports on the use of two dimensional finite element analyses to assess
the impact of soil voids on the deformation of buried rigid pipes after they fracture. Voids
at the springlines and the invert are studied, to determine moment amplification with
increases in the extent of the void. Real soil voids likely have three dimensional geometry,
and ingress of water through joints and longitudinal fractures leads to erosion adjacent to
these features, Figure 1.10. Just as in Chapter Two, this three dimensional geometry is
idealized as a system of equivalent two dimensional soil voids, Figure 1.11, and two
dimensional finite element analyses were employed. Two dimensional analyses were also
conducted to model the laboratory tests reported by Spasojevic et al. (2007), to examine
the role of soil voids on those tests and to evaluate the effectiveness of the finite element
modeling techniques.
Chapter Five summarizes the research procedures, results, and conclusions. A brief
discussion of the value of the proposed design guideline is presented. Recommendations
for future work are then presented, including additional numerical modeling and
laboratory and field testing.
14
Further details of the modeling from Chapter Two are presented in the Appendix.
15
FERENCES
1. Moore, I.D. 2001. Culverts and buried pipelines, chapter 18 of the Geotechnical and
Geoenvironmental Handbook, pp. 541-568, edited by R.K. Rowe, Kluwer Academic
Publishers.
2. Katona, M.G. 1978. Analysis of long-span culverts by the finite element method,
Transportation Research Record, Transportation Research Board, Washington D.C., 678,
59-66.
3. Hoeg, K. 1968. Stresses against underground structural cylinders, J. Soil Mechanics
and Foundation Engineering, ASCE, 94, (4), 833-858.
4. Moore, I.D., 1988. Static response of deeply buried elliptical tubes, Journal of
Geotechnical Engineering, American Society of Civil Engineers, Vol. 114, No. 6, pp.
672-687.
5. McGrath, T.J. and Kurdziel, J.B. 1991. SPDIA method for reinforced concrete pipe
design, J. Transportation Engineering, ASCE, 117, (4), 371-381.
6. Buco1, J., Emeriault, F., Le Gauffre, P., and Kastner, R. 2007. Statistical and 3D
numerical identification of pipe and bedding characteristics responsible for longitudinal
behavior of buried pipe, Villeurbanne, France
7. Buco1, J., Emeriault, F., Le Gauffre, P., and Kastner, R. 2007. Full scale experimental
determination of concrete pipe joint behavior and its modeling, Villeurbanne, France
8. Law, T.C.M., 2004. Behavior of tight fitting flexible pipe liner under earth loads,
thesis for the degree of Doctor of Philosophy, University of Western Ontario, London,
Ontario, Canada
9. Spasojevic, A.D., Mair, R.J. and Gumbel, J.E., 2007. Centrifuge modeling of the
16
effects of soil loading on flexible sewer liners, Géotechnique, Vol. 57, No. 4, pp. 331-341
10. Gumbel, J., Spasojevic, A., and Mair, R. 2003. Centrifuge modeling of soil load
transfer to flexible sewer liners, Proceedings ASCE Pipeline 2003 Conference, Baltimore,
11 pp.
17
Crown
Shoulder
Springline
Haunch
Invert
Figure 1.1 Definition of Terms for the Pipe
Symmetric about CL
θ
Shear stress Radial stress θττ 2sin2= θσσσ 2cos20 +=
Figure 1.2 External Loads Acting Directly on the Circular Pipes (Moore 2001)
18
Figure 1.3 Thrust and Bending Moment at Crown of Pipe
Figure 1.4 Rigid Pipe with Overloading Fractures (ATV-M 143-2, with Permission)
19
a. Undeformed pipe geometry and specific reference points
b. Deformed pipe quadrant under fracture rotation at invert
Figure 1.5 Kinematics of Fractured Rigid Pipe (after Law, 2004)
20
Applied Pressure (kPa)
0 25 50 75 100 125 150
Dia
met
er C
hang
e (m
m)
-12
-8
-4
0
4
8
12
Perc
enta
ge o
f Out
er P
ipe
Dia
met
er (3
70m
m)
-3%
-2%
-1%
0%
1%
2%
3%
Average Laboratory DataTheory - Bonded InterfaceTheory - Full-Slip Interface
Horizontal
Vertical
Figure 1.6 Pipe Deformation Measurements and Theoretical Deformations
Vs. Applied Pressure (after Law, 2004)
Applied Pressure (kPa)
0 25 50 75 100 125 150
Dia
met
er C
hang
e (m
m)
-12
-8
-4
0
4
8
12
Perc
enta
ge o
f Out
er D
iam
eter
(370
mm
)
-3%
-2%
-1%
0%
1%
2%
3%
Average Laboratory DataFull-Slip InterfaceBonded Interface
Horizontal
Vertical
Figure 1.7 Comparison of Finite Element Analysis and Laboratory Measurements (after
Law, 2004)
21
Figure 1.8 Possible Three Dimensional Void Geometry Due to Water Ingress at Joint
Figure 1.9 Plane Strain Idealization of Void Geometry for Preliminary Analysis
Rigid Pipe
Joint
Void
Joint
Void
Rigid Pipe
22
Void
Joint
Fractured Rigid Pipe
Figure 1.10 Possible Three Dimensional Void Geometry Due to Ingress at Joint and through Longitudinal Fractures
Figure 1.11 Plane Strain Idealization of Void Geometry Due to Ingress at Joint and through Longitudinal Fractures
Fractured Rigid Pipe
Joint
Void
23
CHAPTER 2
EFFECT OF BACKFILL EROSION ON MOMENTS IN BURIED RIGID PIPES1
2.1 INTRODUCTION
Soil-pipe interaction research has resolved a variety of buried pipe limit states. Major
advances in the 1960’s and 1970’s have resulted from the development of soil-pipe
interaction solutions that permitted rational calculation of thrust, moment and
deformation (e.g. the closed form solution of Hoeg 1968, and the finite element
procedure of Katona 1978). Stability limit states such as buckling (Moore 1989), local
bending in the profile and around the circumference (Moore and Hu 1995, Dhar et al.
2004), and local buckling (Dhar and Moore 2001) have all been investigated. Based on
these advances, it might seem that the final challenges associated with analysis and
engineering of the buried pipe problem have been resolved.
Now, durability relates to the ability of a pipe to withstand, to a satisfactory degree, the
effects of service conditions (see NRCC 1998). This has generally been interpreted as
the ability of the pipe to resist wear and decay, and investigations to ensure service life
have generally worked to ensure that sufficient pipe material of sufficient durability has
been employed. However, the growing need to repair and replace aging culvert and sewer
infrastructure has made it clear that durability is not simply a function of the long term2
performance of the structural material. Rather, many rigid pipes (i.e. vitrified clay and 1A version of this chapter has been published for publication in the Transportation Research Record, and presented at the Annual Conference of the Transportation Research Board.
24
reinforced concrete) fail as a result of joint leakage, ingress of groundwater, and the
accompanying erosion of the soil envelope adjacent to the pipe. Workers concerned with
pipe repair using liners have raised important questions regarding the impact of those soil
voids on the repaired pipe (e.g. Gumbel et al. 2003).
A program of research at Queen’s University has recently been initiated to study the
physical processes and implications of deterioration of the backfill soil. Computer studies
are being undertaken to design a laboratory test program, and to provide preliminary
guidance on the acceptable limits of soil erosion. The study reported here quantifies the
impact of soil voids on the bending moments in buried rigid pipes. Idealized soil voids
are defined, and two dimensional finite element analyses are used to assess the impact of
those voids on bending moments. Voids at the springlines and the invert are studied, to
determine moment amplification with increases in the extent of the void. While this study
could be used to provide preliminary guidance on the stability of existing structures with
such erosion voids, it will be followed by experimental studies to assess the reliability of
the computer analysis and to determine how soil erosion and void growth control pipe
failure.
2.2 THE RIGID PIPE
A rigid pipe may be defined as one which, under its maximum load, does not deform
sufficiently to produce significant changes in the earth pressures applied by the soil in
which it is laid. Rigid pipes support loads in the ground by virtue of the resistance of the
25
pipe as a ring to bending. For example, design of steel reinforcement in concrete pipe is
undertaken to ensure the pipe can support the expected bending moments.
Now, while pipe deformations in rigid pipes are not sufficient to influence the bending
moments that develop, the role of the envelope of soil surrounding the pipe is
nevertheless critical. The concrete pipe does not simply act as a stiff ring under the action
of vertical forces at crown and invert (the so-called ‘parallel plate’ loading condition).
Rather, earth pressures are distributed across the top and bottom halves of the pipe, and
lateral earth pressures also develop on the sides of the structure. It is actually the
difference between the vertical and horizontal earth pressures that produces bending
moments, and ‘Bedding Factor’ BF is used in reinforced concrete pipe design to quantify
the reductions in bending moment relative to the case of stiff ring under parallel plate
loading. Use of values of BF exceeding 4 for design of pipes surrounded by good
quality backfill is an indication of the substantial reductions in bending moments that
occur if the pipe is placed within an envelope of good quality backfill.
Based on the product data from the manufacturer and former laboratory tests from Law
2004), a rigid pipe of wall thickness t = 44mm and outer diameter OD = 388mm is
studied, to illustrate the effects of void creation on pipe stability. The pipe is specified
with Young’s Modulus of E = 28GPa, a reasonable value for reinforced concrete.
However, increases or decreases in this value have little effect on the bending moments
that are calculated, as long as this modulus remains orders of magnitude higher than the
surrounding soil. The strength of this rigid pipe will be assumed to be defined by a
26
capacity to resist vertical forces under two or three point loading of 23KN/m, which is
shown on Figure 2.1. A zero value of Poisson’s ratio is used, assuming that the presence
of joints releases constraint in the axial direction. Again, however, the use of other values
will have little if any effect on the calculated bending moments.
2.3 NUMERICAL MODELING – CONCRETE PIPE ALONE
Rigid pipe structural properties in the circumferential direction include the effective
elastic modulus of the pipe wall modulus ( ), the cross-section area per unit length of
pipelines ( ) and the second moment of area per unit length (I). For a long plain pipe
of wall thickness
pE
pA
t
tAp = (2.1)
12
3tI = (2.2)
For this rigid pipe with its load capacity under two point (i.e. parallel plate) loading of F
= 23 KN/m, the moment at the crown, Mcr, and the extreme fiber tensile stress at that
location σcr (stress) can be calculated as follows (Moore 2001):
1−= πFrM cr (2.3)
IMt2
=σ (2.4)
Results of these ‘closed form’ calculations are included on Table 2.1.
The response of the rigid pipe has also been examined using the finite element program
27
AFENA (Carter et al. 1980). This calculation was performed to assess whether the
numerical analysis correctly provides the stress and moment in the reinforced concrete
pipe. Pipe properties identical to those reported in Table 2.1 were used.
The mesh was defined for half of the RC pipe, using the symmetry of the problem, Figure
2.2a. Two hundred and forty triangular elements, each with six nodes, were used to model
the right hand half of the RC pipe, which is adequate to capture the local stress and strain.
The calculated distributions of xxσ and yyσ are shown on Figures 2.2b and 2.2c. The
results of compressive stress and bending moment at the crown of the RCP are 3.95 MPa
and 1.30 KN.m/m, respectively. These are almost identical to the results from the closed
form calculation, demonstrating that the finite element procedure is giving correct
solutions. The rest of this article focuses on this pipe when buried, quantifying the impact
of the voids on the bending moments and tensile stresses at the extreme fibers.
2.4 NUMERICAL MODELING - ELASTIC CALCULATIONS FOR VOIDS AT
SPRINGLINES
Figure 2.3 shows the geometries for a series of voids that were defined at the springlines.
While the void is not expected to have a regular shape, void geometries based on circular
shapes were chosen so that the geometry is clear. The voids were defined so that:
Each ‘set’ of three voids has three different diameters; the smallest contacts the springline
over an arc of approximately 30o (actually 29.6o), the next intersects over 60o (actually
28
59.7o), and the largest void intersects the pipe over a 90o arc (actually 90.8o). The angles
provided in parentheses are somewhat different to the target values, since the angle
choices are constrained by the finite number of nodes used around the external pipe
circumference.
Three different void sets are defined, as shown in Figures 2.3a to 2.3c; these allow the
effect of the lateral extent of the voids to be studied (from wide, Figure 2.3a, to narrow,
Figure 2.3c). Void geometry for 3 void sets are shown on Figure 2.4.
Traditionally, theoretical pipe solutions have been developed for two idealized interface
conditions: perfect adhesion of the soil to the structure (the bonded or no-slip interface
condition), and zero adhesion (the full-slip or smooth interface condition), e.g. Hoeg
(1969) and Moore (2001). Since the actual pipe response in the laboratory or in the field
is expected to lie somewhere between these two limits, both are studied and reported
here.
Each analysis was performed considering a surface pressure from overlying soil of
100kPa (equivalent to 5m of typical compacted backfill).
Finite element analysis was used to calculate the stress distribution across the wall of the
rigid pipe, Figures 2.5 to 2.7. These figures show the results of elastic soil-structure
interaction analysis at the crown, the invert and the springline respectively. Elastic-plastic
analysis is considered in a subsequent section.
29
These elastic calculations, performed for the widest void geometries (Figure 2.3a)
indicate how increases in the diameter of the void (or the angle α over which the void
contacts the exterior of the pipe), increase the magnitude of the extreme fiber stresses and
the bending moments at all key locations (the crown, the invert and the springlines). Pipe
burial depth (or the overburden pressure discussed previously) has been chosen so that
factor of safety against tensile cracking is initially about 2. If this rigid pipe has tensile
cracking stress of 4MPa (the value implied by the analysis presented in Table 2.1), it
could be expected to reach that limit state first at the springline, closely followed at the
crown and invert. This would occur when the void approaches the largest size shown in
Figure 2.3a (a void contacting the pipe over an arc somewhat less than 90o).
It is noticed from Figures 2.5 to 2.7, that the circumferential stresses are not linear as we
expect with thin ring theory due to the thick thickness of the pipe, however it has little
effect on the response of the rigid pipe since all the results are almost linear. Figure 2.8
summarizes the percentage increase in extreme fiber tension with growth in the void-pipe
contact angle. These results indicate that the wider cavities (i.e. void set A) produce
higher tensile stresses relative to the very narrow voids (i.e. void set C). However, the
dominant geometrical factor appears to be the angle of void in contact with the RCP.
A comparison between solutions for the two different interface conditions indicates that
the growth of a void generally has more effect on stresses when the interface is rough (or
bonded).
30
2.5 NUMERICAL MODELING - ELASTIC-PLASTIC CALCULATIONS FOR VOIDS
AT SPRINGLINES
The introduction of a cavity in the soil adjacent to the pipe could reasonably be expected
to induce shear failure in the soil, particularly for cases where the backfill is granular and
it has little if any cohesion. The model used for the soil was therefore extended to include
the influence of shear failure. Backfill soil strength was defined using a friction angle φ
of 200 (and small cohesion of 0.002kPa to remove spurious failure of points at zero
stress). Modulus is set to 2MPa and Poisson’s ratio is set to 0.333. These strength and
modulus choices represent what might be expected for very loose soil, to reflect
significant dilation and weakening of the backfill in the vicinity of the void (given that
water is present and erosion is occurring). Figure 2.9 shows the stress comparison at
crown and springline between elastic analysis and elastic-plastic analysis for bonded
interface, and Figure 2.10 for smooth interface.
Finite element analysis was used to calculate the stress distribution across the wall of the
rigid pipe, Figures 2.11 to 2.13. These figures show the results of elastic-plastic
soil-structure interaction analysis at the crown, the invert and the springline respectively.
Figure 2.14 provides a summary of the resulting changes in tensile and compressive
stress at the extreme fibers at the crown of the pipe. A comparison of these results with
those provided in Figure 2.8 makes it clear that the incorporation of shear failure into the
analysis substantially increases the magnitude of stress enhancement associated with void
formation. For the largest voids (with an angular contact over a 90o arc across the
31
springline), the stresses are enhanced over 200%. Therefore, while shear failure does not
significantly change bending stresses or the likelihood of fracture for pipes buried in
continuous (void free) backfill, it appears that shear failure in the soil substantially
enhances the impact of voids on the demand placed on the buried pipe. This tripling of
stress at the extreme fiber should be more than enough to induce tensile fracture in a
typical reinforced concrete pipe buried near its depth limit.
2.6 NUMERICAL MODELING - ELASTIC-PLASTIC CALCULATIONS FOR VOIDS
AT INVERT
The impact of void development under the invert of a buried pipe was investigated using
the mesh geometries shown in Figure 2.15. These lead to changes in circumferential
stresses across the wall of the pipe at the invert, as illustrated in Figure 2.16 to 2.18.
In each of these cases, the effect of the void at the invert is to reverse the sign of the
bending moments that occur at crown, invert and springlines. The removal of the soil
foundation below the pipe is substantially reducing the vertical stresses acting across the
structure, and eventually the void is large enough to lower the vertical stresses carried
across the pipe below the horizontal stresses it is carrying. At this point, the sign of the
bending moments changes. Perhaps the only very significant circumstance where this
would be of concern is for pipes with elliptical reinforcement, which greatly lowers their
ability to resist these moments of opposite sign (since the steel is placed on the
assumption that radius of curvature increases at the crown and invert, and it decreases at
the springlines.
32
For the invert void, it appears that smooth interface response provides more rapid
changes in bending moments and greater stresses at the extreme fiber.
2.7 DISCUSSION AND CONCLUSIONS
This study demonstrates that the development of a void at the springline results in
substantial increases in bending moments at crown, springlines and invert. Elastic
analysis implies that voids stretching around a 90o arc at the springline acts to
approximately double the bending moments and tensile stresses at the extreme fibers. An
elastic-plastic finite element analysis considering backfill of low strength approximately
triples the bending moments and tensile stresses. These analyses suggest that the erosion
voids can certainly bring the pipe to a stability limit state. The longitudinal fractures that
open along the springlines as a result of these moments could then enhance further
erosion.
At first when erosion begins to form a void beside the pipe springline, the increases in
bending moment are modest and they have little impact on the balance between load and
resistance. However, once the void at the springlines grows to a point where stretches
over an arc extending more than 45o, moments begin to accelerate more rapidly. If joint
repair is able to arrest void growth at this level, the analyses imply that the system will
remain stable. The analyses of larger voids imply that these systems are very likely to
have resistance fall below demand.
33
The analysis of voids formed under the invert implies that substantial reductions in
moment result followed the development of moments of opposite sign. This is analogous
to use of geofoam or other compressible materials under the pipe invert, or induced
trench construction (a void over the crown). This would be of major concern for pipes
with elliptical reinforcement, which depend on moments of specific sign at each of the
critical locations (crown, invert and springlines).
All results presented here are based on theoretical calculations. Improvements to the
predictive capacity of the analysis will depend on the development of relevant physical
data. In particular, buried pipe tests featuring voids at the springline would be valuable to
assess the accuracy of the finite element calculations and the definitions of critical
geometrical (e.g. void shape) and material (soil strength) properties.
34
REFERENCES
1. Hoeg, K. 1968. Stresses against underground structural cylinders, J. Soil Mechanics
and Foundation Engineering, ASCE, 94, (4), 833-858.
2. Katona, M.G. 1978. Analysis of long-span culverts by the finite element method,
Transportation Research Record, Transportation Research Board, Washington D.C., 678,
59-66.
3. Moore, I.D. 1989. Elastic buckling of buried flexible tubes - a review of theory and
experiment, Journal of Geotechnical Engineering, American Society of Civil Engineers,
Vol. 115, No. 3, pp. 340-358.
4. Moore, I.D. and Hu, F. 1995. Response of high-density polyethylene pipe in hoop
compression, Transportation Research Record 1514, Design and Performance of
Underground Pipe, pp. 29-36.
5. Dhar, A.S., Moore, I.D. and McGrath, T.J. 2004. Two-dimensional analysis of
thermoplastic culvert deformations and strains, Journal of Geotechnical and
Geoenvironmental Engineering, ASCE, Vol. 130, No. 2, pp. 199-208.
6. Dhar, A.S. and Moore, I.D. 2001. Liner buckling in profiled polyethylene pipes,
Geosynthetics International, Vol. 8, No. 4, pp. 303-326.
7. National Research Council Canada, 1998. Durability and performance of gravity
pipes: a state-of-the-art literature review, Inst. for Research in Construction.
8. Gumbel, J., Spasojevic, A., and Mair, R. 2003. Centrifuge modeling of soil load
transfer to flexible sewer liners, Proceedings ASCE Pipeline 2003 Conference, Baltimore,
11 pp.
9. Law, T.C.M., 2004. Behavior of tight fitting flexible pipe liner under earth loads,
35
thesis for the degree of Doctor of Philosophy, University of Western Ontario, London,
Ontario, Canada
10. Moore, I.D. 2001. Culverts and buried pipelines, chapter 18 of the Geotechnical and
Geoenvironmental Handbook, pp. 541-568, Edited by R.K.Rowe, Kluwer Academic
Publishers.
11. Carter, J.P. and Balaam, N.P. 1980. AFENA - a general finite element algorithm: user
manual, school of Civil and Mining Engineering, University of Sydney, N.S.W. 2006,
Australia.
36
Table 2.1 Calculation Results of the Maximum Bending Moment, Maximum Stress and Maximum Deflection of the RC Pipe with Parallel Plate Load 23KN/m
Classical Solution Finite Element Analyses Solution
EI 1.99E+08 N.mm 1.99E+08 N.mm EA 1232000 N/mm 1232000 N/mm t 44 mm 44 mm D 344 mm 344 mm E 28000 N/mm^2 28000 N/mm^2 I 7098.67 mm^4/mm 7098.67 mm^4/mm r 172 mm 172 mm
PS 262.55 N/mm/mm 262.55 N/mm/mm L 1 mm 1 mm F 23 N/mm 23 N/mm Deflection 0.0877 mm 0.090 mm M ( ) crM 1259 N.mm/mm 1300 N.mm/mm Stress (�) 3.90 MPa 3.95 MPa Strength (s) 4 MPa 4 MPa Deflection Break 0.0900 mm 0.091 mm
Note : F = PS * Deflection (F is the force per unit length)
37
38
Figure 2.1 Vertical Earth Loads at the Pipe Crown and Invert (2 Point Loading and 3 Point Loading)
a. Mesh b. xxσ distribution under line load of 23 MN/m c. yyσ distribution under line load of 23 MN/m
Figure 2.2 Finite Element Analysis of Rigid Pipe Alone (Tension Negative)
39
Pipe Void1 Void2 Void3
Pipe Void1 Void2 Void3
Pipe Void1 Void2 Void3
a. Widest Voids – Void Set A b iate Width – Void Set B c. Narrow Voids –Void Set C
Figure 2.3 Finite Element Models of Buried RC Pipe with Voids at Springline
40
. Intermed
Figure 2.4 Void Geometry for Void Set A, B & C (Unit: mm)
41
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-6
-4
-2
0
2
4
6
-5
-3
-1
1
3
5
Ste
ss s
xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
Out
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ce
Insi
de S
u rfa
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-6
-4
-2
0
2
4
6
-5
-3
-1
1
3
5
Ste
ss s
xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
Out
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Insi
de S
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a. Bonded interface b. Smooth interface Figure 2.5 Stress xxσ at Crown of the Rigid Pipe – Void Set A – Elastic (Compress Positive)
42
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-6
-4
-2
0
2
4
6
-5
-3
-1
1
3
5
Ste
ss s
xx (M
Pa)
No VoidRemove void1Remove void2Remove void3
Out
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-6
-4
-2
0
2
4
6
-5
-3
-1
1
3
5
Ste
ss s
xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded interface b. Smooth interface Figure 2.6 Stress xxσ at Invert of the Rigid Pipe – Void Set A - Elastic (Compress Positive)
43
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-4
-2
0
2
4
6
-5
-3
-1
1
3
5
7
Stes
s s
xx (M
Pa)
No voidRemove void1Remove void2Remove void3
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-4
-2
0
2
4
6
-5
-3
-1
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3
5
7
Ste
ss s
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Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded interface b. Smooth interface Figure 2.7 Stress yyσ at Springline of the Rigid Pipe – Void Set A – Elastic (Compress Positive)
44
-125
-100
-75
-50
-25
0
25
50
75
100
125
0 25 50 75
Void Contact Angle ( ° )
Perc
enta
ge C
hang
e of
Stre
sses
(%
100
)
Void Set A (Comp. Stress - Bonded Condition) Void Set B (Comp. Stress - Bonded Condition)Void Set C (Comp. Stress - Bonded Condition) Void Set A (Comp. Stress - Smooth Condition)Void Set B (Comp. Stress - Smooth Condition) Void Set C (Comp. Stress - Smooth Condition)Void Set A (Tension Stress - Bonded Condition) Void Set B (Tension Stress - Bonded Condition)Void Set C (Tension Stress - Bonded Condition) Void Set A (Tension Stress - Smooth Condition)Void Set B (Tension Stress - Smooth Condition) Void Set C (Tension Stress - Smooth Condition)
Figure 2.8 Elastic Analysis of Increases in Extreme Fiber Stresses at the Crown as a Function of Void Geometry and Interface Condition
45
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
0 25 50 75
Void Contact Angle ( ° )
Stre
sses
(MPa
) - C
ompr
ess P
ositi
v
100
e
Void Set A (Stress @ Crown - E) Void Set B (Stress @ Crown - E)
Void Set C (Stress @ Crown - E) Void Set A (Stress @ Crown - EP)
Void Set B (Stress @ Crown - EP) Void Set C (Stress @ Crown - EP)
Void Set A (Stress @ Springline - E) Void Set B (Stress @ Springline - E)
Void Set C (Stress @ Springline - E) Void Set A (Stress @ Springline - EP)
Void Set B (Stress @ Springline - EP) Void Set C (Stress @ Springline - EP)
Figure 2.9 Stress Comparison (@ Crown and Springline) between Elastic Analysis and Elastic- Plastic Analysis for Bonded Interface (E – Elastic, EP – Elastic-Plastic)
46
47
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 25 50 75 100
Void Contact Angle ( ° )
Stre
sses
(MPa
) - C
ompr
ess P
ositi
v
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
e
Void Set A (Stress @ Crown - E) Void Set B (Stress @ Crown - E)
Void Set C (Stress @ Crown - E) Void Set A (Stress @ Crown - EP)
Void Set B (Stress @ Crown - EP) Void Set C (Stress @ Crown - EP)
Void Set A (Stress @ Springline - E) Void Set B (Stress @ Springline - E)
Void Set C (Stress @ Springline - E) Void Set A (Stress @ Springline - EP)
Void Set B (Stress @ Springline - EP) Void Set C (Stress @ Springline - EP)
Figure 2.10 Stress Comparison (@ Crown and Springline) between Elastic Analysis and
Elastic- Plastic Analysis for Smooth Interface (E – Elastic, EP – Elastic-Plastic)
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-6
-4
-2
0
2
4
6
-5
-3
-1
1
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Ste
ss s
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Pa)
No voidRemove void1Remove void2Remove void3
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-6
-4
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2
4
6
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Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded interface b. Smooth interface Figure 2.11 Stress xxσ at Crown of the Rigid Pipe – Void Set A – Elastic-Plastic (Compress Positive)
48
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-6
-4
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0
2
4
6
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3
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Pa)
No voidRemove void1Remove void2Remove void3
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-6
-4
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2
4
6
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-3
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3
5
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ss s
xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded interface b. Smooth interface Figure 2.12 Stress xxσ at Invert of the Rigid Pipe – Void Set A - Elastic-Plastic (Compress Positive)
49
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-4
-2
0
2
4
6
-5
-3
-1
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3
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7
Stes
s s
xx (M
Pa)
No voidRemove void1Remove void2Remove void3
Out
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-4
-2
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4
6
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3
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7
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xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded interface b. Smooth interface Figure 2.13 Stress yyσ at Springline of the Rigid Pipe - Void Set A - Elastic-Plastic (Compress Positive)
50
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 25 50 75
Void Contact Angle ( ° )
Perc
enta
ge C
hang
e of
Stre
sses
(%
100
)
Void Set A (Comp. Stress - Bonded Condition) Void Set B (Comp. Stress - Bonded Condition)Void Set C (Comp. Stress - Bonded Condition) Void Set A (Comp. Stress - Smooth Condition)Void Set B (Comp. Stress - Smooth Condition) Void Set C (Comp. Stress - Smooth Condition)Void Set A (Tension Stress - Bonded Condition) Void Set B (Tension Stress - Bonded Condition)Void Set C (Tension Stress - Bonded Condition) Void Set A (Tension Stress - Smooth Condition)Void Set B (Tension Stress - Smooth Condition) Void Set C (Tension Stress - Smooth Condition)
Figure 2.14 Elastic-plastic Analysis of Increases in Extreme Fiber Stresses at the Crown
as a Function of Void Geometry and Interface Condition
51
52
Pipe Void3 Void2 Void1
Figure 2.15 Mesh Defining Voids under Invert (Void Set D)
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-6
-4
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a. Bonded Interface b. Smooth Interface Figure 2.16 Stress xxσ at Crown of the Rigid Pipe – Void Set D – Elastic-Plastic (Compress Positive)
53
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-6
-4
-2
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6
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Pa)
No VoidRemove Void1Remove Void2Remove Void3
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-6
-4
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2
4
6
-5
-3
-1
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3
5
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ss s
xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded Interface b. Smooth Interface Figure 2.17 Stress xxσ at Invert of the Rigid Pipe – Void Set D – Elastic-Plastic (Compress Positive)
54
-20 -10 0 10 20-15 -5 5 15Distance from the middle of the pipe (mm)
-4
-2
0
2
4
6
-5
-3
-1
1
3
5
7
Ste
ss s
xx (M
Pa)
No VoidRemove Void 1Remove Void2Remove Void3
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-4
-2
0
2
4
6
-5
-3
-1
1
3
5
7
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ss s
xx (M
Pa)
No VoidRemove Void1Remove Void2Remove Void3
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a. Bonded Interface b. Smooth Interface Figure 2.18 Stress yyσ at Springline of the Rigid Pipe – Void Set D – Elastic-Plastic (Compress Positive)
55
CHAPTER 3
NUMERICAL STUDY OF EFFECT OF THICKNESS ON THE DEFORMATION OF FRACTURED PIPE: RESPONSE TO EARTH
LOADS1
3.1 INTRODUCTION
Full scale laboratory analysis, which investigated the kinematic response of an
artificially fractured rigid sewer pipe, was undertaken together with finite element
analysis of the fractured sewer by Law and Moore (2002a). That study indicates that
the changes in vertical and horizontal pipe diameter of a rigid pipe with overloading
fractures are similar to a flexible pipe, in that deformation is controlled by the
surrounding soil. While there is some additional bending stiffness within the pipe
segments, the full release of moment at crown, invert and springlines appears
sufficient to ensure that the changes in vertical and horizontal pipe diameter are very
similar to those for flexible pipe. They concluded that bonded idealized flexible pipe
theory using secant soil modulus provided very effective calculations for fractured
pipe deformations due to changes in earth loads. However, Law and Moore (2002a)
did not conduct a general parametric study to quantify the effect of rigid pipe
thickness on those deformations. This chapter presents finite element analyses that
reveal the relationship between deformation of a fractured rigid pipe and its wall
thickness.
The characteristics of rigid pipe were defined in Chapter Two. Rigid pipe fracture
configurations and deformation mechanisms were also explained by Law (2004), and
T1 A version of this paper has been submitted for publication as a technical note in the ASCE Journal of Geotechnical and Geoenvironmental Engineering.
56
typical deformations are illustrated in Figures 3.1 and 3.2. The fractures in the rigid
pipe expand at the inner surface of the pipe at the crown and invert, and the outer
surface at the springlines. One contact point is maintained at each fracture during
deformation, forming a hinge point. The flexural response of fractured pipes is
expected to be governed by plane strain conditions, where the fractures run parallel to
the pipe axis.
3.2 DESCRIPTION OF THE FINITE ELEMENT MODELING
The finite element program, AFENA, was used to study the effect of earth loads on
the behavior of buried rigid pipes with different thicknesses. The primary objective is
to conduct analysis to capture the critical features of the buried fractured pipe-soil
interaction. The boundary condition between the pipe and the soil may significantly
influence the pipe response. Traditionally, theoretical pipe solutions (e.g. Hoeg 1968,
Burns and Richard 1964) have been developed for two idealized interface conditions:
perfect adhesion of the soil to the structure (the bonded or no-slip interface condition),
and zero adhesion (the full-slip or smooth interface condition). Since the actual pipe
response in the laboratory or in the field is expected to lie somewhere between these
two limits, both are studied here (more complex modeling of the interface featuring
friction was not subsequently conducted based on the success of these limiting cases).
Models were established, denoted MA1, for use in the numerical analysis, Figure 3.3.
To calculate the pipe deformations, two-dimensional plane strain analyses were
carried out. The finite element analyses were undertaken to check the effect of the
location of the side boundary show that rigid pipe deflection only increased by 0.8%
when mesh width was doubled, so the location of the side boundary has negligible
57
affect on results. Only half of the system was analyzed due to the symmetry of the
problem. Close to nine hundred six-node triangles was used to model the soil, and
finer elements were used in the vicinity of the pipe in order to get more accurate
results.
The young’s modulus of the reinforced concrete pipe is 3.3 GPa, and has no effect on
the calculated response, since it is essentially rigid compared to soil. Joint elements,
0.25mm in length, were placed between the pipe and the soil to model the interface,
which is an arbitrary choice to separate the nodes. For the bonded pipe-soil condition,
these joint elements are prescribed with both normal and shear stiffness. Conversely,
high normal stiffness but negligible shear stiffness was specified for the full-slip
condition.
3.3 MODELING PIPES WITH DIFFERENT THICKNESS
Model MA1 features a rigid pipe with thickness of 44mm (t/OD = 44/388 = 11.3%,
Figure 3.4a) made up of four rings of elements. A series of other fractured pipe
deformation calculations were performed with this mesh by adjusting the location of
the ‘hinge’ (or segment contact) point at the pipe springline, placing it at distances of
33mm (t/OD = 33/388 = 8.5%, Figure 3.4b), 22mm (t/OD = 22/388 = 5.7%, Figure
3.4c), 11mm (t/OD = 11/388 = 2.8%, Figure 3.4c), and finally 0mm from the soil-pipe
boundary. This produced fractured pipe deflection calculations for this range of
thickness to diameter ratios (or dimension ratios DR of 8.8, 11.8, 17.6, 35.3 and an
indeterminately high value). In these calculations, the ‘inactive’ part of the pipe (that
at radial coordinates less than the hinge position) was left in place, and played no part
58
in affecting the interaction of the segments of pipe or the surrounding ground (other
than lend stiffness to the pipe segments during calculation, so that they had negligible
shape change or distortion. A check calculation was performed with these elements
removed, and this confirmed that they had no influence on the response to the model,
and then to maintain pipe segment rigidity for the case t=0). Shown on Figure 3.8, t
and has the same results for the finite element analyses. eqt
While this representation of the pipe thickness and the hinge point is based on the
assumption that segments of rigid pipe of thickness will be in contact at the inside
surface where they meet at the springlines, the possibility exists that the pipe
segments will crush where they contact at the springline, so that the hinge point
moves some distance out towards the soil-structure interface, Falter (1980). In this
case, the results would be interpreted based on an equivalent thickness, i.e. the
distance along the fracture from the revised hinge point to the boundary with the soil,
Figure 3.6.
3.4 SOIL MODELING AND LOADING SEQUENCE
Soil type considered in the analysis was a low stiffness material with Young’s
modulus of 2 MPa, Poisson’s ratio of 0.333, angle of internal friction φ’ of 20o, and
dilation angle of 20o. If the dilation angle is changed to 10°, the result will be changed
less than 1%, so it has a small effect on pipe deformation. The models were built up
gradually by placing the soil in layers up until full loading. All analysis assumes
‘drained’ conditions, where the loading is applied gradually. To ensure the
elastic-plastic analysis of the normally consolidated material is numerically stable, a
59
small cohesion values was assigned to the clay.
The sequence of loading history and pipe deterioration steps was as follows:
L2: Set up the finite element model where the burial depth of the rigid pipe is 806mm
from the outside of the pipe at the crown to the top surface of the mesh.
A2: Articulation (release the fractures in the rigid pipe).
N2: Apply additional vertical pressure at the top of the soil in 10 increments of 10 kPa
up to a full loading of 100 kPa.
During the finite element analysis of MA1, a large stress concentration was found in
the soil adjacent to the pipe springlines. To explain this, consider points S1 and S2 in
contact with the soil above and below the fracture, Figure 3.6. These separate as the
fracture opens (the pipe segments remain in contact and rotate about the inner point of
the fracture, S3). Soil elements E1, E2 and E3 in contact with S1 and S2 are subjected to
large vertical strains that would not occur in practice, since the joint separation would
not be imposed on a single point in the soil.
To remove this artificial stress concentration, model MA2 was prepared based on the
same mesh, dimensions and analysis sequence at MA1, but with the critical soil
element E2 and a few joints between soil and pipe removed, Figure 3.7. The resulting
stress distributions no longer featured the artificial stress concentration, and it was
confirmed that it did not introduce other numerical problems (for example, shear
failure was not triggering in the soil).
Mentioned in Section 3.2 and shown on Figure 3.8, the finite element analyses were 2 Notation “L”, “A” and “N” refers to the burial, articular (Fracture) and additional load application phases of the analysis.
60
undertaken to check the effect of equivalent thickness of rigid pipe shows that the two
different models have the same results, which means that t equals to . eqt
The effect of the unrealistic circumferential constraint at the springlines that occurs
with model MA1 is further discussed in the next section.
3.5 RESULTS
Figures 3.9 and 3.10 present the results for bonded and full-slip conditions
respectively, and loading stage N of finite element model MA1 with different
thickness ratios t/OD (as discussed in 3.3 above). Increases in horizontal diameter
appear on the top half of these figures, and the decreases in vertical diameter on the
bottom half. Direct comparisons of results for bonded and full-slip interface
conditions are shown on Figure 3.11.
Figures 3.12 and 3.13 present the results obtained using model MA2. A direct
comparison of results for bonded interface and full-slip interface is presented in
Figure 3.15.
The analyses using MA1 reveal that, regardless of interface condition, a pipe with
negligible thickness (0) has vertical diameter decrease equal in magnitude to the
horizontal diameter increase between springlines. For wall thickness which is 5.7% of
OD, the vertical diameter decrease is about 88% of the horizontal diameter change.
For wall thickness of 11.4%, the vertical diameter change decreases to 78% of the
change in horizontal diameter. In each case, this reflects the kinematics of pipe
61
deformation, Law and Moore (2002b), where the radial distances to the points of
rotation at crown-invert and the springlines results in:
)/21(/ ODtDD hv −−≈∆∆ (3.1)
This estimate of diameter change ratio is exact when the fractured sewer begins to
deform, and provides values that are somewhat high as deflections increase. At
diameter change ratios of 10% (where the values described above were calculated),
Equation 3-1 produces estimates within 1% of the actual ratio.
Analyses for MA1 indicate that there is a dramatic increase in pipe deformations as
pipe thickness decreases, both for bonded and full-slip interfaces. However, this is a
reflection of the stress concentration discussed in 3.3 above, which represents restraint
at the springlines that is an artificial result of the analysis.
For MA2, the analyses again show that a pipe with negligible thickness has vertical
and horizontal diameter changes of equal magnitude. For pipes with finite thickness,
the ratios of hv DD ∆∆ / are the same as those obtained using model MA1. Results
for full-slip interface are somewhat higher than those for bonded interface. The
dramatic decreases in horizontal deformations with thickness increase seen in
calculations using model MA1 are not seen in Figures 3.12, 3.13, and 3.15. There are
only small decreases in horizontal diameter change as thickness rises from 0% to 22%.
There are decreases in the magnitude of vertical diameter change, but these appear
almost completely the result of the kinematics expressed in Equation 3.1. Shown on
Figure 3.14, the results from Figure 3.13 for vertical deformations are compared with
the idealized flexible pipe theory times by (1 - 2t/OD), which further prove that the
results are very close.
62
The theoretical response of an idealized flexible pipe (high hoop stiffness, negligible
bending stiffness, perfectly bonded pipe-soil interface, Moore 2001) is
∆Dh = - ∆Dv = 8 σv R (1-K0) (1-νs) (1+νs)/ [(3-2νs) Es] (3.2)
where
σv = vertical earth pressures
K0 = coefficient of lateral earth pressure
R = OD/2 = radius of the pipe-soil boundary
Es = Young’s modulus of the soil
νs = Poisson’s ratio of the soil
Calculations using Equation (3.2) are also shown on Figures 3.9, 3.10, 3.12 and 3.13.
At 100 kPa, the change in diameter of the idealized flexible pipe is about 9.3%.
Results from the finite element analyses indicate that the changes in horizontal pipe
diameter of a rigid pipe with overloading fractures is almost identical to a flexible
pipe, where deformation is controlled by the surrounding soil and thickness has little
influence on deformation. While there is significant bending stiffness within the pipe
segments, the full release of moment at crown, invert and springlines appears
sufficient to ensure that the pipe response is largely the same as flexible pipe.
Bonded idealized flexible pipe theory using secant soil modulus is very effective in
estimating horizontal diameter increasing of fractured pipe deformations due to
changes in earth loads.
3.6 IMPLICATION FOR LINER DESIGN
63
Law and Moore (2003) demonstrate that a liner inserted within a sewer fractured into
four quadrants is subjected to local bending stresses at crown and invert. Local
bending strains can be calculated from the decrease in vertical pipe diameter as:
22
139.21
82 liner
v
liner
vincr R
cD
R
cD ∆±=
⎟⎠⎞
⎜⎝⎛ −
∆±==
πππ
εε (3.3)
where
εcr = tensile bending strain on the outside of the liner at the crown
εin = tensile bending strain on the outside of the liner at the springline
c = distance from the neutral axis of the liner to the outside liner surface (t/2 for a
plain liner)
Rliner= average radius of the liner
Since the finite element analysis has demonstrated that Equation (3.1) effectively
provides the ratio between magnitudes of the changes in vertical and horizontal pipe
diameter, and that changes in horizontal diameter can be effectively estimated using
flexible pipe theory, liner stains can be estimated by combining Equation (3.2) and
Equation (3.3) to become:
2
0
)23(
)1)(1)(1(4)21(139.2
linerss
ssv
incr RE
KODOD
tc
ν
ννσεε
−
+−−−== (3.4)
The finite element analyses have demonstrated that response along the fractured
pipe-soil interface is closer to the full-slip condition than the bonded one. However,
the use of bonded flexible pipe theory appears to be able to capture the deformations
of the fractured pipe generally well. For a fractured pipe, bending stiffness is
64
released between segments (across the fractures), but is retained within segments. A
flexible pipe, on the other hand, has negligible bending stiffness around its entire
circumference. The release of bending stiffness at the four fractures appears
sufficient to bring the overall response of the damaged rigid pipe to a condition
equivalent to a buried flexible pipe.
3.7 SUMMARY AND CONCLUSION
Finite element analysis has been used to investigate the flexural behavior of fractured
sewer pipes with different thicknesses. The analysis demonstrates that rigid pipe
thickness has essentially no effect on the changes in horizontal pipe diameter that
occur when a fracture pipe deforms under the influence of earth pressures. While the
fractured rigid pipe has bending stiffness within intact pipe segments, the full release
of moment at the fractures at crown, invert and springlines appears sufficient to
ensure that the changes in vertical and horizontal pipe diameter are similar to those for
flexible pipe. Bonded idealized flexible pipe theory (Moore 2001) was found to be
very effective in calculating the increase in horizontal diameter of the fractured pipe
due to changes in earth loads. The finite element analysis demonstrated that the
decrease in vertical diameter can be scaled from the increase in horizontal diameter
using the simple relationship (1 - 2t/OD) obtained from considerations of the fractured
pipe kinematics. A simple modification based on this multiplication factor is proposed,
correcting the liner design model of Law and Moore (2002b) to account for pipe
thickness.
65
REFERENCES
1. Law, T.C.M. & Moore, I.D. 2002a. Laboratory investigation on the static response
of repaired sewers, proceeding of the ASCE 2002 Pipeline Conference, Cleveland,
Ohio.
2. Law, T.C.M. 2004. Behavior of tight fitting flexible pipe liner under earth loads,
thesis for the degree of Doctor of Philosophy, University of Western Ontario, London,
Ontario, Canada.
3. Hoeg, K. 1968. Stress against underground cylinder, Journal of Soil Mechanics
and Foundation Engineering, ASCE, Vol. 94, SM4, pp 833-858.
4. Burns, J.Q. and Richard, R.M. 1964. Attenuation of Stresses for buried cylinders,
proceeding of the symposium on soil-structure interaction, University of Arizona,
379-392.
5. Law T.C.M. and Moore, I.D. 2002b. Kinematic response of damaged rigid sewer
under earth load, proceeding of annual conference, Canadian Geotechnical Society,
Niagara Falls, ON, October.
6. Moore, I.D. 2001. Culverts and buried pipelines, chapter 18 of the Geotechnical
and Geoenvironmental Handbook, pp. 541-568, edited by R.K.Rowe, Kluwer
Academic Publishers.
7. Law, T.C.M. and Moore, I.D. 2003. Guidance on design of flexible liners to
repair structurally compromised gravity flow sewers, No Dig Conference, Las Vegas,
March 30 - April 3, 2003.
66
Rigid Pipe Fractures
Figure 3.1 Idealized Deformation of Damaged Pipes – before Disturbance (after Law and Moore, 2004)
Hinge at Contact Points
Angular Expansion θ
Figure 3.2 Idealized Deformation of Damaged Pipes – after Disturbance (after Law and Moore, 2004)
67
Figure 3.3 Finite Element Model of Buried Reinforced Concrete Pipe (MA1 & MA2)
68
(a) Pipe thickness (44mm) (b) Pipe Thickness (33mm)
Figure 3.4 Finite Element Model Details of Buried Reinforced Concrete Pipe (MA1 & MA2)
69
(c) Pipe Thickness (22mm) (d) Pipe thickness (11mm)
Figure 3.5 Finite Element Model Details of Buried Reinforced Concrete Pipe (MA1 & MA2)
70
Figure 3.6 Detail “A”
71
Figure 3.7 Detail “B”
72
73
Backfill Soil
Backfill Soil
Pipe Segments
a
Pipe Overlap when Pipe Deforms at Hinge ‘a’
Joints Modeling Fractures
eqt
Figure 3.8 Detail “C” ( )eqtt =
t
Pipe Segments
Joints Modeling Fractures
0 40 8020 60 100Applied Pressure (kPa)
-40
-20
0
20
40
-30
-10
10
30P
ipe
Def
orm
atio
n (m
m)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Per
cent
age
of P
ipe
Out
er D
iam
eter
(388
mm
) (%
)
t/D=11.3% (Bonded)t/D=8.5% (Bonded)t/D=5.7% (Bonded)t/D=2.8% (Bonded)t/D=0.0% (Bonded)Idealized FlexiblePipe Theory
Vertical
Horizontal
Figure 3.9 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied
Pressure – Bonded Interface
0 40 8020 60 100Applied Pressure (kPa)
-40
-20
0
20
40
-30
-10
10
30
Pip
e D
efor
mat
ion
(mm
)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Per
cent
age
of P
ipe
Out
er D
iam
eter
(388
mm
) (%
)
t/D=11.3% (Full-Slip)t/D=8.5% (Full-Slip)t/D=5.7% (Full-Slip)t/D=2.8% (Full-Slip)t/D=0.0% (Full-Slip)Idealized Flexible Pipe Theory
Vertical
Horizontal
Figure 3.10 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Full-Slip Interface
74
0 40 8020 60 100Applied Pressure (kPa)
-40
-20
0
20
40
-30
-10
10
30P
ipe
Def
orm
atio
n (m
m)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Per
cent
age
of P
ipe
Out
er D
iam
eter
(388
mm
) (%
)
t/D=11.3%t/D=8.5% t/D=5.7% t/D=2.8%t/D=0.0%
Vertical
Horizontal
BondedFull-Slip
Figure 3.11 Horizontal and Vertical Deformation of Rigid Pipe Comparison between
Full-Slip Interface and Bonded Interface
-40
-20
0
20
40
-30
-10
10
30
Pip
e D
efor
mat
ion
(mm
)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Per
cent
age
of P
ipe
Out
er D
iam
eter
(388
mm
) (%
)
t/D=11.3% (Bonded)t/D=8.5% (Bonded)t/D=5.7% (Bonded)t/D=2.8% (Bonded)t/D=0.0% (Bonded)Idealized Flexible Pipe Theory
0 40 8020 60 100Applied Pressure (kPa)
Vertical
Horizontal
Figure 3.12 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied
Pressure – Bonded Interface
75
0 40 8020 60 100Applied Pressure (kPa)
-40
-20
0
20
40
-30
-10
10
30P
ipe
Def
orm
atio
n (m
m)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Per
cent
age
of P
ipe
Out
er D
iam
eter
(388
mm
) (%
)
t/D=11.3% (Full-Slip)t/D=8.5% (Full-Slip)t/D=5.7% (Full-Slip)t/D=2.8% (Full-Slip)t/D=0.0% (Full-Slip)Idealized Flexible Pipe Theory
Vertical
Horizontal
Figure 3.13 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Applied
Pressure – Full-Slip Interface
0 40 8020 60 100
Applied Pressure (kPa)
-40
-30
-20
-10
0
-35
-25
-15
-5
Pip
e D
efor
mat
ion
(mm
)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Per
cent
age
of P
ipe
Out
er D
iam
eter
(388
mm
) (%
)
t/D=11.3% (Full-Slip)t/D=8.5% (Full-Slip)t/D=5.7% (Full-Slip)t/D=2.8% (Full-Slip)t/D=0.0% (Full-Slip)Idealized Flexible Pipe Theory * (1-2t/D), t/D=11.3%Idealized Flexible Pipe Theory * (1-2t/D), t/D=8.5%Idealized Flexible Pipe Theory * (1-2t/D), t/D=5.7%Idealized Flexible Pipe Theory * (1-2t/D), t/D=2.8%Idealized Flexible Pipe Theory * (1-2t/D), t/D=0.0%
Vertical
Figure 3.14 Results Compassion between Idealized Flexible Pipe Theory and
Numerical Modeling Vertical Deformation of Rigid Pipe with Increasing Applied Pressure – Full-Slip Interface
76
0 40 8020 60 100Applied Pressure (kPa)
-40
-20
0
20
40
-30
-10
10
30P
ipe
Def
orm
atio
n (m
m)
-10.0
-5.0
0.0
5.0
10.0
-7.5
-2.5
2.5
7.5
Perc
enta
ge o
f Pip
e O
uter
Dia
met
er (3
88m
m) (
%)
t/D=11.3%t/D=8.5% t/D=5.7% t/D=2.8%t/D=0.0%
Vertical
Horizontal
BondedFull-Slip
Figure 3.15 Horizontal and Vertical Deformation of Rigid Pipe Comparison between
Full-Slip Interface and Bonded Interface
77
CHAPTER 4
NUMERICAL STUDY OF DEFORMATIONS IN FRACTURED
SEWERS: RESPONSE TO EARTH LOADS AND EROSION VOIDS1
4.1 INTRODUCTION
Chapter 2 presented finite element results quantifying the effect of backfill erosion on
moments in buried rigid pipe, and their potential to fracture. Computer analysis is used
here to study the deformations that occur after the rigid pipe fractures. The effect of
erosion voids in the soil at the springlines and invert are examined.
Centrifuge tests on soil load transfer to flexible sewer liners have been reported by
Spasojevic et al. (2007). They showed that void formation is the dominant factor
affecting the soil-pipe interaction; it produced 60% to 70% of the total amount of liner
distortion. They also conclude that void collapse adjacent to the springline has a greater
effect on liner distortion than voids formed under the invert. Their test configuration is
studied here, to examine whether the computational analysis is able to capture the
observed behavior.
All calculations reported here are based on pipe-soil interaction modeled using the finite
element program AFENA (Carter et al. 1980).
4.2 EFFECT OF EROSION VOIDS ON FRACTURED PIPE DEFORMATIONS
1 A version of this paper has been submitted for publication to the Geotechnique.
78
4.2.1 Introduction to The Finite Element Modeling
Numerical modeling of fractured rigid pipes under earth loads and the behavior of tight
fitting flexible sewer liners has been developed and discussed by Law and Moore (2002),
and this problem was studied in the previous chapter to establish the effect of wall
thickness on the pipe deflections. However, no work has been performed, to date, to
study the effect of erosion voids in the vicinity of the pipe on fractured pipe deformations.
Numerical modeling is therefore developed in this section to explore the deformations
that result as voids develop.
Finite element program, AFENA, was again used and all definitions, boundary conditions
and interface conditions are the same as those described in Chapter 3. The primary
objective is to conduct finite element analysis to capture the critical elements of the
buried fractured pipe-soil interaction. In order to analyze the effects of the models with
different voids, models (MB) were established for various void characteristics. The
meshes used in this numerical study are shown in Figure 4.2.
In the absence of forensic information providing details of three dimensional voids that
develop in the field (likely with greater size in the vicinity of leaking joints which are
initially responsible for groundwater inflow and backfill erosion), a first approximation of
the void geometry as prismatic is employed. This uniform cross sectional void geometry
along the pipe axis means that plane strain analysis can be used. Idealized geometry of
erosion voids are shown on Figure 4.1.
79
All calculations presented in this section are for a rigid pipe with wall thickness of 44mm
and outer diameter of 388mm. While this thickness of typical of North American concrete
pipes of this diameter, the study reported in Chapter 3 indicates that vertical diameter
change in a fractured rigid pipe is influenced by pipe thickness, and it can be calculated
from horizontal diameter change values using a correction factor of OD
t21− (t is the
wall thickness, and OD is the outer diameter).
4.2.2 Modeling of Soil and Erosion Voids
An elastic plastic material was again used to represent the soil, with parameters chosen to
represent a typical clay backfill. Properties are the same as those described in Chapter 3,
except that two combinations of Poisson’s ratio sν and coefficient of lateral earth
pressure K were investigated: 333.0=sv (K=0.5) and 4.0=sv (K=0.667). Both elastic
analysis (where high cohesion values are employed so that no shear failure occurs) and
elastic-plastic analysis were employed, so that these results can be discussed and
compared. Specifically, the soil is assigned modulus Es of 2 MPa, angle of internal
friction �’ of 20o, dilation angle of 20o, and cohesion c’ of 0.002 MPa. If the dilation
angle is changed to 10°, the result will be changed less than 1%, so it has a small effect
on pipe deformation. The interface between soil and pipe is modeled as either bonded or
smooth. The bonded condition involves full transfer of shear stress and normal stress and
full compatibility of displacements between pipe and ground. The smooth interface
condition involves ‘full slip’ between the pipe and soil; while normal stresses are
80
transferred across the interface, shear stresses are not.
Model MB features six different voids, denoted MBV1, MBV2, MBV3, MBV4, MVB5,
and MBV6, each with specific contact angles and void depths, as illustrated on Figure 4.2.
The total cross sectional areas of each of these voids are reported in Table 4.1.
4.2.3 Analysis Sequence
Two different loading histories were considered, to examine how the point where pipe
fracture occurs influences the fractured pipe deformations. Loading and deterioration
history are represented by a sequence of steps. The sequence of those steps for the MB1
analyses was as following (Table 4.1):
L: Set up the finite element model, with a burial depth to the crown of the rigid pipe of
806mm, the self weight of the soil elements in the mesh above the rigid pipe is not
considered in the analysis.
A: Articulation, where the fractures are ‘introduced’ by removing all but one of the joint
elements at each fracture, so that flexural ring stiffness is released across the fractures
N: Vertical pressure is applied at the top of the soil model in ten increments of 10 kPa up
to a maximum surface loading of 100 kPa; there are still no voids introduced into the soil
at the end of this stage.
V1: The first void MBV1 is excavated
V2: The second void MBV2 is excavated
81
V3: The third void MBV3 is excavated
V4: The fourth void MBV4 is excavated
V5: The fifth void MBV5 is excavated
V6: The sixth void MBV6 is excavated
The loading and deterioration steps for analysis sequence MB2 are the same as MB1,
except that articulation occurs after excavation V2: (Table 4.1):
L – N – V1 – V2 – A – V3 – V4 – V5 – V6
This is closer to the load path expected in the field, where the pipe fractures result
because of void formation.
4.2.4 Results
Results of horizontal and vertical diameter changes are normalized as)1(.
.KD
ED
v
s
−∆σ
, using
Es, the Young’s Modulus of the soil, D, the outer diameter of the undamaged pipe, vσ , the
total vertical stress at the crown of the pipe, and lateral earth pressure coefficient K.
Results were calculated for K=0.5 and K=0.667, and for both bonded and smooth
soil-pipe interface conditions. Results are presented in Figure 4.3 relative to the size of
the void, quantified using the contact angle between pipe and voids at the springline.
Results for model MB2, where fractures (articulation) occurs after excavation V2, are
given on Figure 4.4, using contact angle to define the size of the void.
82
4.2.5 Discussion
For both the elastic and elastic-plastic finite element analyses, the calculated results for
the bonded interface are very similar to those for smooth pipe-soil interface, so it appears
that interface condition has little influence on the deformations that result (even before
excavation reduces the length of contact between soil and pipe).
For elastic analysis, the calculated results are the same at 45°for K=0.5 and K=0.667.
When the angular contact is less than 45°, the normalized deformations for coefficient of
lateral earth pressure K=0.667 are smaller than those for K=0.5. When the angular
contact is larger than 45° (void set VB3 is removed), the results for K=0.667 are larger
than those for K=0.5.
For elastic-plastic analysis, the low cohesion leads to shear failure during voids removal.
Deformations then increase dramatically when the angle of contact between pipe and
voids (i.e. void area) increases. When the contact angle exceeds 60°, the damaged rigid
pipe has deformed substantially.
Finite element analysis MB2 shows that before the rigid pipe fractures, deformations are
small (they can be neglected). However after pipe fracture, the deformations increase
dramatically, and become exactly those seen for model MB1. It does not appear
necessary to consider the exact point where fractures developed to calculate the rigid pipe
deformations after fracture. Naturally, this simplifies the calculation of fractured pipe
83
deformations. It also simplifies the possible use of closed circuit television images to
interpret soil condition from pipe ovality (as discussed in the next section).
Earlier research by Law and Moore (2002, 2004) reveals that the deformation of repaired
sewers involves interaction between three components: the soil, the fractured pipe, and
the liner. The finite element analysis undertaken here demonstrates that the size of the
erosion voids can have a substantial influence on the local strains that develop at the
crown and invert of a liner, as the fractured rigid pipe surrounding that liner is deformed.
4.2.6 Interpretation of Pipe Deflections to Infer ‘Damaged’ Soil Condition
One issue facing engineers who design pipe repair systems is the likely extent of
disturbance to the surrounding soil as a result of erosion, and the long term impact of soil
deterioration on polymer liner stability. The finite element calculations presented here
indicate that the size of a void besides the pipe directly influences the deformations that
the sewer will experience. There is a possibility of using the shape of damaged sewers
observed from Closed Circuit Television (CCTV) images to assess the likely extent of
soil voids. The results presented here show that there is not one unique relationship
between deformations and void geometry. This relationship depends on the value of
Poisson’s ratio and the coefficient of lateral earth pressure, as well as the level of shear
failure in the surrounding ground.
Two approaches could be used to interpret observations of diameter change. First, it is
84
possible to use flexible pipe theory and interpret a value of ‘equivalent’ soil modulus.
This theory considers the soil to be of uniform condition (without voids), so the modulus
is reduced below its original undisturbed value to characterize the stiffness of the
remaining soil and the voids. Alternatively, it might be possible to define a ‘nominal’ void
size based on an assumption of elastic or elastic-plastic soil response. Generally it is
easiest to estimate changes in horizontal diameter from CCTV images, since the invert is
often obscured by wastewater. Using Figures 4.5, a horizontal diameter
increase)1(.
.KD
ED
v
s
−∆σ
, of 3, for example, could be interpreted in two manners:
• First, this could be considered a result of reduction in ‘uniform’ (zero void) soil
modulus by between 2 and 3, e.g. Moore (2001); Figure 4.3 indicates that)1(.
.KD
ED
v
s
−∆σ
ranges between 1.0 and 1.5 for K value between 0.5 and 0.67.
• Alternatively, a void in ‘elastic’ soil would produce that level of deformation when
void contact angle at each springline lies between 55o and 65o.
If the surrounding soil had the strength used in the elastic-plastic analyses presented here,
these interpretations would change. For example, void angle would instead be assessed as
being between 30o and 40o.
More work is needed to explore the use of these approaches as a means of characterizing
the damaged soil-rigid pipe system, including physical testing to better establish the
shape of erosion voids, and the effectiveness of the analytical methods that have been
developed.
85
4.3 FINITE ELEMENT ANALYSIS OF CENTRIFUGE TESTS EXAMINING SOIL
LOAD TRANSFER TO FLEXIBLE SEWER LINER.
4.3.1 Description of The Centrifuge Modeling
Spasojevic et al. (2007) conducted a series of centrifuge modeling tests of soil load
transfer to a flexible liner placed within a ‘rigid’ pipe, buried within uniform granular
backfill. Figures 4.6 and 4.7 provide a plan view and cross-section of the centrifuge
package developed to model the problem. The tests were run at a centrifugal acceleration
of 30g, so the pipe of nominal OD 100mm (actual 101.6mm) at burial depth of 150mm
simulates a prototype sewer of 3m OD with 4.5m of cover, all other dimensions being in
the same proportion.
For his laboratory tests, the host pipe was composed of four hinged quadrants of Dural
tube. The pipe hinges were designed to simulate deterioration of the host pipe in two
steps while under earth pressures resulting from 30 times normal gravity (30g):
(1) “Articulation”: where the cap restraining the end of the pipe (initially keeping it in
its original circular geometry) is removed to activate the 4-hinge mechanism,
effectively eliminating the flexural ring stiffness of the host pipe (simulating the
effect of longitudinal cracks at crown, springlines and invert); and
(2) “Circumferential shortening”: where tapered pins used in each of the hinges are
extracted, so that the pin becomes loose within the hole, and it reduces the
circumference of the host pipe.
86
The flexible liner was modeled using commercial PVC pipe of OD (outside diameter) of
75mm, leaving a nominal annular gap of 0.6mm, when it was inserted in the host pipe.
Soil voids in the vicinity of the test pipe were simulated using rubber bladders mounted
over mandrels running parallel to the pipe axis. These were located either below the
invert or on either side of the springlines. While these were inflated with incompressible
fluid at the start of the test (during soil placement and gravitational acceleration), the
collapse of the soil voids was then simulated by staged release of fluid from one or
multiple bladders, at controlled flow rates and with continuous monitoring of pressure,
via a solenoid valve, stand pipe and linear hydraulic actuator. Soil used in all the tests was
Leighton Buzzard sand (grade E),which was ‘rained’ in and around the other components
from a fixed height to achieve a consistent density of 1500 kg/m3 in all tests.
Finite element program AFENA has been used to model the Spasojevic et al. (2007) tests.
This numerical procedure includes explicit representation of the liner, the hinged host
pipe, with the objective of assessing and interpreting the test measurements. One half of
the soil-host pipe-liner system was modeled given the symmetry of the problem, model
geometry is shown in Figure 4.8. The numerical analysis commenced with simulation
of centrifuge acceleration of 30g. The analysis incorporated modeling of both host pipe
deterioration (the articulation and circumferential shortening phases), as well as void
formation in the surrounding soil.
4.3.2 Modeling of Host-Pipe Liner System Information
87
a. Details of the numerical modeling
Around two hundred six-noded triangular elements were used to model the right hand
half of the host pipe (two 90°segments). Young’s modulus of the host pipe was set as 70
GPa, and it was given density of . The wall thickness of the host pipe was
12.7mm, and the outer diameter (OD) was 101.6mm. This model host pipe tested at 30g
simulates a prototype pipe of outer diameter 3.05m and wall thickness 381mm. While
this represents a very large rigid pipe (a diameter larger than normally seen in practice),
the results should still prove informative, providing guidance on rigid pipe response
following deterioration.
3/2600 mkg
The outer diameter (OD) of the flexible liner modeled in the finite element analysis was
75mm, and liner thickness (t) was 1.8 mm. The Young’s modulus of the liner material as
measured was 2.758 GPa and unit mass of the liner material was . Based on the
internal diameter of the host pipe (76.2 mm) and the external diameter of the liner (75.0
mm), the nominal annular gap between the host pipe and liner in the analysis was 0.6mm.
Forty six-node triangular elements were used to model half of the liner (two 90°
segments).
3/900 mkg
b. Modeling of soil and void information
The soil used in all the finite element analysis was sand with Young’s modulus of 4.2
88
MPa and unit mass as . Two mesh zones were used to model the soil: an inner
zone near the pipe and including the voids, using fine six-node triangular elements, and
an outer layer, with larger element size, Figure 4.9.
3/1500 mkg
Voids with initial diameter of 34mm were modeled at the same level as the springline,
separated from the pipe by the distance reported by Spasojevic et al. (2007). Similarly,
voids were modeled the correct distance under the invert, Figure 4.9. Ninety eight
six-node triangular elements were used to model the void at the springline. Forty eight
six-node triangular elements were used to model the half void at the invert. To match the
experimental geometry, the voids were modeled as being over 12.7 mm in diameter,
running parallel to the pipe axis either below the invert or on either side of the spring line.
Pressure was maintained at the required overburden pressure during gravitational
acceleration, which models behavior with the voids filled with soil.
c. Modeling of Joints
The previous numerical studies in Chapter 3 & 4 on fractured pipe deformation (for intact
and eroded backfill, respectively), indicate that the interface conditions has a small effect
on pipe deformation (much smaller than other factors, such as Poisson’s ratio for the soil
or void size). Therefore, only one interface condition was examined. Subsequently,
evenly distributed joints with high normal stiffness and very low shear stiffness were
used to model a smooth interface between soil and pipe, and between liner and pipe.
These were assigned joint stiffness properties to ensure there was no shear stress between
89
soil and pipe. To model the pipe fracture at the springline, invert and crown, a small gap
was left between the pipe segments and joints with high normal stiffness and high shear
stiffness were defined between the different mesh segments. To model articulation, the
pipe segments in the centrifuge model are tied by stiff springs so that they maintain
contact at the middle of the pipe wall (the physical model used in the centrifuge had
hinges located at these positions). The remaining joint elements were prescribed with
negligible normal and shear stiffness to allow the fracture to deform without any
resistance. The joints with negligible stiffness have no effect on the calculation results,
and could have been removed. At the circumferential shorting stage, all the joints
between pipe segments were allowed to deform freely.
4.3.3 Numerical Modeling Sequence
The numerical modeling was conducted using a sequence of steps. The loading and
deterioration steps used to model the standard loading cycle used in the centrifuge
modeling were as follows:
G (Gravity): Spin the centrifuge test package up to acceleration of 30g.
A (Articulation): This step transforms the sound rigid host pipe into an articulated system
of four quadrants; it is induced by being initially restrained with an end plug, so that there
is rotation between the pipe segments.
V1 (First stage void formation): Elements representing 10% of total void soil were
excavated (deactivated), either at springline or invert depends on the void location for
90
that analysis.
C (Circumferential shorting): This stage modeled the effects of circumferential
compression (shortening) of the host pipe, which means joints were partially extracted to
leave a gap between the pipe segments.
V2 (Second stage void formation): Further soil elements were excavated (removed) to
eliminate 25% of the total soil area in the void.
V3 (Third stage void formation): All remaining elements in the void were removed; bring
it to 100% soil of the total void area.
For the case of partial void collapse, Figure 4.10 shows the elements that were removed,
namely a zone of elements closest to the pipe springline.
4.3.4 Results and Discussion
A total of two numerical analyses were undertaken as listed in Table 4.2. The results of
the calculated changes in vertical and horizontal diameter are shown on Figure 4.11,
through the various stages of analyses HS1 and HI1.
It is can be seen that the calculated changes in vertical and horizontal pipe diameter are
essentially equal and opposite throughout. The initial phase G3 of the numerical
modeling analysis produced a negligible distortion for both two models. At stage A, both
numerical solutions produced the same distortions of 2.4% ( Dh /δ or Dv /δ ). After Stage
V3, numerical analyses of HS1 produce similar vertical and horizontal diameter changes
of about 6%, and analyses of HI1 produce values of about -0.2%. The negative sign used
91
here means the change in the horizontal diameter was negative (it shortened), while the
vertical diameter stretched (the opposite of what was observed during the test).
The results show that void formation at the springline is the dominant factor affecting
deformations, and the soil-pipe interaction. In each analysis it produced about 95% of the
total amount of the liner distortion for voids at that location. However, when the voids
were located under the invert, the analysis indicates that collapse of voids leads to
vertical diameter extension, and horizontal diameter contraction (the opposite of what has
been observed in the experiments).
A comparison of the laboratory measurements of Spasojevic et al. (2007) with the finite
element results for Model HS1 are shown on Figure 4.12. The figure indicates that many
phases of the liner deformations for the laboratory test and the numerical analysis are
similar. Now, to assist with a more detailed comparison, the different sections of the
deformation histories have been labeled a-b-c-d-e-f-g-h-i and a’-b’-c’-d’-e’-f’-g’-h’-i’ to
distinguish between the measured and calculated responses, respectively.
i. Segments a-b and a’-b’ (Phase G) are distinctly different. The computer analysis
gives essentially zero deformation since the pipe is not yet modeled as fractured
(the hinges have not yet been released). Measurements indicate over 1% diameter
change (positive for the horizontal pipe diameter, and negative for the horizontal
pipe diameter). It appears that the restraint provided at the hinges on the test pipe
was not complete, and that some rotation was permitted at these points.
92
ii. Segments b-c-d and b’-c’-d’ are also distinctly different (phase A). The finite
element analysis features modeling of a sudden release of the hinges (rapid
fracture development over b’-c’). The test structure appears to have experienced
further ongoing deformation in this part of the ‘articulation’ phase. By the end of
the articulation phase A, the deformation of the test structure (point d) is about
40% higher than the calculated deformation (d’).
iii. Additional deformation accumulated over phase V1 (segments d-e and d’-e’) are
very similar for the test pipe and the numerical model.
iv. Additional deformations over the first part of phase C is similar (e-f versus e’-f’),
but there is a sudden change in the measured deformation at point f, so that the
measured diameter change at the end of this part of the laboratory test (g) has
crept closer to the calculated value (g’). Possibly there is restraint in the physical
model that does not occur in the numerical idealization, but part of that restraint
was suddenly removed at point f.
v. During soil removal stage V2, the diameter change accumulated in the experiment
(g to h) exceeds that seen in the numerical model (g’ to h’), though this brings
points h and h’ into close proximity. The removal of the continuum elements used
in the void region during phase V2 brings the experimental and numerical models
into almost identical configurations, perhaps reflecting use of fluid within the
‘voids’ installed in the physical model. These may have a significant effect on the
the earth pressures and pipe deformations prior to that point.
vi. Stage V3, has measured deformations h-i that are very different to calculated
values (h’-i’). Likely the test procedure involved removal of high g values during
93
this phase of the test, and these sections will not be examined further.
Figure 4.13 shows a comparison between the deformations from the physical and
numerical models for HI1 (where the void is located at the invert). While responses
through the G and A phases are similar to those discussed earlier in relation to Figure
4.12, but the patterns of deformation once the void starts to be ‘excavated’ (phase V1 and
beyond) are dramatically different. While the numerical analysis indicates that a void
located under the pipe changes the sense of deformation (it is equivalent to placing the
rigid pipe on a compressive foundation like styrene, so that lateral stresses exceed vertical
stresses, and the incremental pipe deformations are in the pattern of a vertical ellipse).
The physical model does not exhibit this behaviour, and it appears that there may be
substantial complexities in the manner in which the physical model is behaving.
4.4 SUMMARY AND CONCLUSION
A numerical study of the deformation of buried rigid pipes (with or without liner)
deforming as a result of longitudinal fractures has been presented. Deformation
enhancement due to erosion voids in the backfill has been studied, using finite element
analysis. This first study of this phenomenon has employed idealized void geometry, with
location adjacent to the pipe springlines. Elastic and elastic-plastic finite element analyses
reveal that deformation of the damaged rigid pipe increases dramatically with void
growth, accelerating as the erosion void develops a large angle of contact with the pipe.
Contact angle appears to be the dominant factor affecting broken pipe deformations. The
finite element analysis also demonstrates that fractured pipe deflections are independent
94
of the point at which the fractures occurred (that is, diameter changes are essentially the
same whether the fractures preceded void erosion, or whether the pipe fractured after the
voids started to form).
These solutions are based on a specific set of idealized pipe damage and void geometries,
and material conditions. Following evaluation using physical models, it may be possible
to use CCTV images of pipe deformation to characterize the condition of deteriorated soil
using an ‘equivalent’ soil modulus reduced to account for ground loss, or some form of
nominal void size.
Finite element analysis was conducted of centrifuge model tests, examining soil load
transfer to flexible sewer liners as sewers fracture, and as voids form in the vicinity of the
sewer. The finite element models produced diameter changes in the ‘fractured’ pipe after
voids were fully formed were very similar in magnitude to those observed in the tests.
However, the manner in which the physical model of the pipe used for the centrifuge tests
was temporarily restrained, and the use of fluids within ‘soil’ zones to be removed during
testing to model ‘erosion’, appears to have lead to significant differences during a number
of the intermediate steps. The effect of voids located under the invert of the pipe was also
dramatically different, and it appears that the laboratory test featured physical
characteristics that are not understood or not modeled in the finite element analysis (such
as movement of soil from beside the test pipe into the void under the pipe invert).
95
ACKNOWLEDGEMENTS
The experimental tests of centrifuge modeling of soil load transfer to flexible sewer liners
were carried by Dr. Aleksandar D Spasojevic (Cambridge University), Prof. Robert J Mair
and Dr. John E Gumbel.
REFERENCES
1. Spasojevic, A.D., Mair, R.J. and Gumbel, J.E. 2007. Centrifuge modeling of the effects
of soil loading on flexible sewer liners, Géotechnique, Vol. 57, No. 4, pp. 331-341.
2. Carter, J.P. and Balaam, N.P. 1980. AFENA - a general finite element algorithm: user
manual, school of Civil and Mining Engineering, University of Sydney, N.S.W. 2006,
Australia.
3. Law, T.C.M. & Moore, I.D. 2002. Laboratory investigation on the static response of
repaired sewers, proceeding of the ASCE 2002 Pipeline Conference, Cleveland, Ohio.
4. Law, T.C.M. 2004. Behavior of tight fitting flexible pipe liner under earth loads, thesis
for the degree of Doctor of Philosophy, University of Western Ontario, London, Ontario,
Canada
5. Gumbel, J.E. 1998. Structural design of pipe linings 1998 – review of pinciples, practice
and current developments worldwide, proceeding of the 4th National Conference on
Trenchless Technology, Brisbane, Australia, August, pp225-239.
6. Moore, I.D. 2001. Culverts and buried pipelines. Chapter 18 of the Geotechnical and
Geoenvironmental Handbook, pp. 541-568, Edited by R.K.Rowe, Kluwer Academic
Publishers.
96
Table 4.1 List of Different Finite Element Models (MB1 & MB2)
Model Name Analysis
Sequence
Void
Information
Void Name Areas
( ) 2mm
Angle
(o )
MBV1 1008.57 15.80
MBV2 4141.85 29.58
MBV3 9562.40 44.52
MBV4 17435.58 59.66
MBV5 27931.17 75.08
MB1 – Elastic-
Plastic Analysis
and voids at
spring line
L – A - N – V1
– V2 – V3 – V4
– V5 – V6
Void shape refer
to Figure 4.1
MBV6 41225.21 90.84
MBV1 1008.57 15.80
MBV2 4141.85 29.58
MBV3 9562.40 44.52
MBV4 17435.58 59.66
MBV5 27931.17 75.08
MB2 – Elastic-
Plastic Analysis
and voids at
spring line
L – N – V1 –
V2 – A – V3 –
V4 – V5 – V6
Void shape refer
to Figure 4.1
MBV6 41225.21 90.84
Table 4.2 List of Different Finite Element Models (HS1 & HI1)
Model Name Analysis Sequence Void Position
HS1 G1 - G2 - G3 - A - V1 - C - V2 - V3
Springline
HI1 G1 - G2 - G3 - A - V1 - C - V2 - V3
Invert
97
Figure 4.1 Finite Element Model of Buried Reinforced Concrete Pipe with Voids at Springline (MB)
98
Figure 4.2 Finite Element Mesh Shows Details of Buried Reinforced Concrete Pipe with Voids at Springline (MB); Contact Angle Illustrated for void MBV4
99
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
0 10 20 30 40 50 60 70 80 90 100
Void Contact Angle (°)
(∆D
* E
s) /
(D*σ
v*(1
-K))
(Hor
izon
tal D
ef. P
ositi
ve, V
ertic
al D
ef. N
egat
ive)
K=0.5, Bonded Interface, Elastic Analysis K=0.5, Smooth Interface, Elastic Analysis
K=0.5, Bonded Interface, Elastci-Plastic K=0.5, Smooth Interface, Elastic-Plastic
K=0.667, Bonded Interface, Elastic Analysis K=0.667, Smooth Interface, Elastic AnalysisK=0.667, Bonded Interface, Elastic-Plastic K=0.667, Smooth Interface, Elastic-Plastic
Figure 4.3 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Angular Contact between Pipe and Voids at Springline (MB1)
100
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
0 10 20 30 40 50 60 70 80 90 100
Void Contact Angle (°)
(∆D
* E
s) /
(D*σ
v*(1
-K))
(Hor
izon
tal D
ef. P
ositi
ve, V
ertic
al D
ef. N
egat
ive)
K=0.5, Bonded Interface, Elastic Analysis K=0.5, Smooth Interface, Elastic Analysis
K=0.5, Bonded Interface, Elastci-Plastic K=0.5, Smooth Interface, Elastic-Plastic
K=0.667, Bonded Interface, Elastic Analysis K=0.667, Smooth Interface, Elastic AnalysisK=0.667, Bonded Interface, Elastic-Plastic K=0.667, Smooth Interface, Elastic-Plastic
Figure 4.4 Horizontal and Vertical Deformation of Rigid Pipe with Increasing Angular Contact between Pipe and Voids at Springline (MB2)
101
Figure 4.5 Estimation of Changes in Horizontal Diameter – Option 2
102
Figure 4.6 Plan Layout of the Centrifuge Package, Spasojevic et al. (2007)
Figure 4.7 Cross-section of the Model, Spasojevic et al. (2007)
103
Figure 4.8 The Numerical Model of the Pipe-Liner-Soil System
104
Figure 4.9 Host Pipe-Liner-Soil Finite Element Mesh
105
Figure 4.10 Elements in Parts of the Void Removed, Namely a Zone of Elements
Closest to Pipe Springline
106
Figure 4.11 Analysis of Diameter Changes for Models HS1 and HI1
107
Figure 4.12 Results Comparison between Lab Test and Numerical Method Analysis (Model HS1)
108
Figure 4.13 Results Comparison between Lab Test and Numerical Method Analysis (Model HI1)
109
CHAPTER 5 DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS
5.1 DISCUSSION AND CONCLUSIONS
Research examining the effect of deterioration of the backfill soil around buried rigid
pipes has been presented. The purpose of the study is to identify and quantifying bending
moment changes with void growth developed in the backfill soil beside and under the
pipe; explaining and quantifying the effects of wall thickness on the pipe deformations,
quantifying how deflection develops with further growth of voids after fractures develop.
Preliminary guidance was given on the acceptable limits of soil erosion based on the
computer modeling. The elastic and elastic-plastic analyses demonstrate that the
development of a void at the springline results in substantial increases in bending
moments at crown, springlines and invert; these erosion voids can certainly bring the pipe
to a stability limit state. When erosion voids are small (contacting the exterior of the pipe
over an angle less than 30 degrees), they have little impact on the balance between load
and resistance. However, when voids enlarge (angle of contact exceeding 45 degrees),
moments begin to accelerate rapidly. Thus if void growth can be arrested at an early stage,
the pipe-soil system should remain stable.
Based on the numerical models with the acceptable limits of soil erosion voids before
110
rigid pipe facture, another numerical study was developed to study the deformation of
fractured buried rigid pipes (with or without liner) with erosion voids in the vicinity.
Elastic and elastic-plastic finite element analyses both reveal that deformation of the
damaged rigid pipe can increase dramatically due to the erosion voids, and the contact
angle is the dominant factor affecting deformation of the broken rigid pipe. The finite
element analysis also shows that pipe deflections for a specific void size are essentially
unaffected by the point in void growth at which pipe fracture occurred (that is, fractured
pipe deflections at specific void size are the same regardless of whether the fractures
occurred early or late during the void growth process).
Law and Moore (2002) developed a theoretical model to correlate the unique relationship
between the vertical and horizontal diameter changes of the fractured rigid pipe under
overloading factures. They demonstrated that the vertical and horizontal diameter
changes are equal and opposite when the fractured pipe has thickness close to zero, and
that the fractured pipe behavior is similar to an idealized flexible pipe. The current
investigation further verified the conclusions mentioned above, and also provided the
relationship between fractured rigid pipe deformations and pipe wall thickness. The
analysis demonstrates that rigid pipe thickness has essentially no effect on the changes in
horizontal pipe diameter that occur when a fracture pipe deforms under the influence of
earth pressures. The finite element analysis also demonstrated that the decrease in vertical
diameter can be scaled from the increase in horizontal diameter using the simple
111
relationship (1 - 2t/OD) obtained from considerations of the fractured pipe kinematics. A
simple modification based on this multiplication factor is proposed, correcting the liner
design model of Law and Moore (2002) to account for pipe thickness.
Finite element analysis was conducted of centrifuge model tests based on a series of
physical experiments reported by Spasojevic et al. (2007), to exam soil load transfer to
flexible sewer liners when sewers fracture, and as voids form in the vicinity of the sewer.
Results show that when the voids are located at springline, the computer study and
laboratory observations are similar, though some differences occur at certain stages.
However when the voids are located at the invert, the computer study suggests that
deformations are of opposite sign; this is very different to the laboratory data. It appears
that there are physical phenomena occurring in the experiments that are not modeled in
the computer analysis; further numerical and physical modeling is needed to study these
and other effects of erosion voids developing near fractured rigid pipe.
Above all, we conclude that deterioration of the backfill soil around buried rigid pipe will
affect the stability and durability of underground sewer system. How to control the
deterioration will be a big issue for engineering design in the future.
5.2 RECOMMENDATIONS
112
All results of the thesis are based on theoretical calculations. A comprehensive laboratory
study should be undertaken to complement the research presented in this thesis.
Ideally, laboratory data capturing the characteristics of idealized erosion voids in the soil,
and other conditions after rigid pipe fracture was encountered should be compared with
the conclusions of this research. Laboratory studies could be used to control all the
relevant parameters; examine the propagation of increasing bending moment and voids
expansion; and further prove the relationship between the increasing deflection and void
growth after pipe fracture.
Prototype scale laboratory experiments to investigate the kinematic response of an
artificially fractured rigid sewer pipe were undertaken by Law and Moore. Further
laboratory work is needed to evaluate the conclusion that decrease in vertical diameter
can be scaled from the increase in horizontal diameter using the simple relationship (1 -
2t/OD), obtained from considerations of the fractured pipe kinematics.
Finally, since it is likely that void growth often starts adjacent to joints, the geometry of
the voids may well be three dimensional. Once there physical evidence is available
regarding the geometry of these voids, a three dimensional finite element study could be
valuable to explore the impact of these geometrical details.
113
REFERENCES
1. Law, T.C.M. & Moore, I.D. 2002. Laboratory investigation on the static response of
repaired sewers, proceeding of the ASCE 2002 Pipeline Conference, Cleveland, Ohio.
2. Law, T.C.M. 2004. Behavior of tight fitting flexible pipe liner under earth loads, thesis
for the degree of Doctor of Philosophy, University of Western Ontario, London, Ontario,
Canada
3. Moore, I.D. 2001. Culverts and buried pipelines. Chapter 18 of the Geotechnical and
Geoenvironmental Handbook, pp. 541-568, Edited by R.K.Rowe, Kluwer Academic
Publishers.
4. Spasojevic, A.D., Mair, R.J. and Gumbel, J.E. 2007. Centrifuge modeling of the effects
of soil loading on flexible sewer liners, Géotechnique, Vol. 57, No. 4, pp. 331-341.
5. Gumbel, J., Spasojevic, A., and Mair, R. 2003. Centrifuge modeling of soil load
transfer to flexible sewer liners, Proceedings ASCE Pipeline 2003 Conference, Baltimore,
11 pp.
114
APPENDIX A
FINITE ELEMENT ANALYSIS OF BURIED RC PIPE TEST – ELASTIC ANALYSIS (VOIDS AT SPRINGLINE)
Finite element analysis has recently been initiated to study the physical processes and
implications of deterioration of the backfill soil. The analysis here quantifies the impact
of soil voids on the bending moments in buried rigid pipes. Idealized soil voids are
defined, and two dimensional finite element analyses are used to assess the impact of
those voids on bending moments. Voids at the springlines and the invert are studied, to
determine moment amplification with increases in the extent of the void. Introduction,
analysis sequence and conclusion have been detailed in Chapter Two. More details about
finite element elastic analysis of buried rigid pipe (voids at springlines) are presented
here.
A.1 DESCRIPTION OF THE FINITE ELEMENT MODELING
The following models are used to study the behaviors of buried sewer RC rigid pipes
with erosion void in the vicinity in the soil. The finite element program, AFENA, was
used to study the earth load effect on the behavior of buried rigid pipe with erosion voids
in the vicinity. The primary objective is to conduct finite element analysis to capture the
critical element of the buried pipe-soil interaction. The boundary condition between the
pipe and the soil may significantly influence the pipe response. Traditionally, theoretical
pipe solution have been developed for two idealized interface conditions: perfect
adhesion of the soil to the structure (the bonded or no-slip interface condition), and zero
115
adhesion (the full-slip or smooth interface condition). Since the actual pipe response in
the laboratory or in the field is expected to lie somewhere between these two limits, both
are studied here.
To calculate the pipe model, two-dimensional plane strain analysis was carried out. The
two dimensional plane boundary condition is a reasonably representation of the strain
condition in the stress cell because of the stiff side walls and the large sample length to
diameter ratio. In order to analyze the effects of the models with different voids
information, we set up 3 different models – MC1, MC2 and MC31 – with different voids
information (Table A.1). Only half the biaxial test cell was analyzed due to symmetry of
the problem. The interface between the soil and the RC pipe was modeled as smooth and
bonded. Close to eleven hundred six-node triangles were used to model the backfill soil,
and finer elements were used in the vicinity of the pipe.
The modulus of the RC pipe has no effect on the calculated response, since it is
essentially rigid compared to soil. Joint elements, 0.25mm in length, were placed between
the pipe and the soil to model the interface, and it has negligible effect on the results. For
the bonded pipe-soil condition, these joint elements are prescribed with both high normal
and shear stiffness. Conversely, high normal stiffness but negligible shear stiffness was
specified for the full-slip condition. The stiffness value of the joints with high stiffness
was chosen so that the joints are stiff enough and slipping does not occur. Additional
analyses using different stiffness values were performed to ensure numerically stability
1 MC1, MC2 and MC3 correspond to Void Set A, Void Set B and Void Set C in Chapter Two.
116
and convergence of the result. The vertical pressure was applied at the top of the soil
using of 100 KPa.
A.2 MODELING OF PIPE AND SOIL INFORMATION
The pipe properties was modeled the same as the pipe introduced in Chapter 2. The soil
used in all analyses was clay with the young’s modulus E = 2 MPa, passion ratio r =
0.333 and friction angle θ = . The vertical pressure applied at the top of the soil
models was built up gradually by increment of 10 KPa until full loading (100 KPa). In
order to make sure the finite element analyses are elastic, the cohesion of the soil was set
to a high value and no shear failure would happen during the test.
°20
A.3 MODELING OF EROSION VOIDS INFORMATION
Model MC1 has different voids which are MC1V1, MC1V2 and MC1V3. Model MC2
has different voids which are MC2V1, MC2V2 and MC2V3 and Model MC3 has
different voids which are MC3V1, MC3V2 and MC3V3, more details refer to Table A.1.
In order to find the change of bending moment and stresses of the RC pipe with different
void, we assume that MC1V1, MC2V1 and MC3V1 have the same angular angle from
the center of the rigid pipe, which 1α = . Same as MC1V2, MC2V2 and MC3V2
have the angular angle from the center of the rigid pipe
°6.29
2α = and MC1V3,
MC2V3 and MC3V3 have the angular angle
°7.59
3α = . °8.90
117
A.4 TEST SEQUENCE
The loading history in the test was represented by a sequence of loading and removing
void steps. The sequence was as following:
L: Set up the finite element model, the buried depth of the RC rigid pipe is 806mm from
the outside of the pipe at crown to the top of the soil.
N: The vertical pressure applied at the top of the soil models was built up gradually by
increment of 10 KPa until full loading (100 KPa); there are no voids at the end of the
stage. (No Void)
V1: After the vertical pressure was applied to the model, the first voids were removed, in
3 different models, MC1V1 and MC2V1 and MC3V1 were removed separately. (Remove
Void1)
V2: The second voids were removed after the first voids removed, in 3 different models,
MC1V2 and MC2V2 and MC3V2 were removed separately. (Remove Void2)
V3: The third voids were removed after the second voids removed, in 3 different models,
MC1V3 and MC2V3 and MC3V3 were removed separately. (Remove Void3)
A.5 RESULTS
From theoretical analysis above, the maximum compressive stress, maximum tensile
stresses and maximum bending moment will occurred at the critical sections of the RC
pipe, which are at Crown, Invert and Springlines.
118
Model MC1
The results of xxσ at Crown of the RC pipe are shown on Figure 2.5a (bonded
conditions) and 2.5b (smooth condition). The results of xxσ at invert of the RC pipe
are shown on Figure 2.6a (bonded conditions) and 2.6b (smooth condition). The results of
yyσ at springline of the RC pipe are shown on Figure 2.7a (bonded conditions) and 2.7b
(smooth condition). The results of xxσ distribution of soil are shown from Figure A.1 to
Figure A.4; the results of yyσ distribution of soil are shown from Figure A.5 to Figure
A.8.
The results of xxσ at Crown of the RC pipe are also shown on Table A.2 and the ratio
changes are shown on Table A.3. The results of xxσ at invert of the RC pipe are shown
on Table A.4 and the ratio changes are shown on Table A.5. The results of yyσ at
springline of the RC pipe are shown on Table A.6 and the ratio changes are shown on
Table A.7.
Model MC2
The results of xxσ at Crown of the RC pipe are shown on Table A.8 and the ratio
changes are shown on Table A.9. The results of xxσ at invert of the RC pipe are shown
on Table A.10 and the ratio changes are shown on Table A.11. The results of yyσ at
springlines of the RC pipe are shown on Table A.12 and the ratio changes are shown on
Table A.13.
119
Model MC3
The results of xxσ at Crown of the RC pipe are shown on Table A.14 and the ratio
changes are shown on Table A.15. The results of xxσ at invert of the RC pipe are shown
on Table A.16 and the ratio changes are shown on Table A.17. The results of yyσ at
springline of the RC pipe are shown on Table A.18 and the ratio changes are shown on
Table A.19.
A.5 RESULTS ANALYSIS
The results shown from Table A.3 to A.19 show that when the pipe and soil are perfectly
adhered together (bonded case), the calculated increasing rate of maximum compressive
stress, maximum tensile stresses and maximum bending moment at critical positions is
higher than the full slip case. Increasing rate of maximum compressive stress, maximum
tensile stresses and maximum bending moment are similar at crown and invert positions.
There is a higher increasing rate at springline compared to crown and invert positions.
The results of comparison of compressive stress and tensile stress of RC pipe at crown of
different models (MC1, MC2 and MC3) are shown on Figure 2.8. For contact angle 90.8°,
the compressive stress of RC pipe @ crown of MC1 (bonded interface) increased 106.0%,
and the compressive stress of RC pipe @ crown of MC1 (full-slip interface) increased
64.3%. For contact angle 90.8°, the tensile stress of RC pipe @ crown of MC1 (bonded
interface) increased 122.3%, and the tensile stress of RC pipe @ crown of MC1 (full-slip
interface) increased 91.7%. Therefore, erosion voids would affect more for bonded
120
121
interface than full-slip pipe-soil interface. It also proves that the contact angle is critical
parameter related to the void information, which increase the stress and moments of the
pipes.
Increasing ratio of compressive stress and tensile stress of RC Pipe with increasing areas
at certain angles for different pipe-soil interfaces are shown from Table A.20 to A.25. The
biggest increasing ratios of each table are marked by grey background. For the worst case,
at contact angle 90.8°, when the void area increased 282.2%, the compressive stress of
RC pipe @ crown of MC1 (full-slip interface) increased 19.0%. Therefore, the void area
is not very important compared to the contact angle when considering the design of
buried pipe system.
a. Bonded interface b. Smooth interface
Figure A.1 Stress xxσ of Soil – MC1 (No Void)
122
a. Bonded interface b. Smooth interface
Figure A.2 Stress xxσ of Soil – MC1 (Remove Void1)
123
a. Bonded interface b. Smooth interface
Figure A.3 Stress xxσ of Soil – MC1 (Remove Void2)
124
a. Bonded interface b. Smooth interface
Figure A.4 Stress xxσ of Soil – MC1 (Remove Void3)
125
a. Bonded interface b. Smooth interface
Figure A.5 Stress yyσ of Soil – MC1 (No Void)
126
a. Bonded interface b. Smooth interface
Figure A.6 Stress yyσ of Soil – MC1 (Remove Void1)
127
a. Bonded interface b. Smooth interface
Figure A.7 Stress yyσ of Soil – MC1 (Remove Void2)
128
a. Bonded interface b. Smooth interface
Figure A.8 Stress yyσ of Soil – MC1 (Remove Void3)
129
Table A.1 List of Different Finite Element Models
Model Name Test Sequence Void Information Void Name Areas
( ) 2mm
Angle ( ) o
MC1V1 4141.9 29.6 MC1V2 17435.6 59.7
Void shape refer to Figure 2.3a & Figure 2.4 MC1V3 41225.2 90.8
MC2V1 1272.7 29.6 MC2V2 8221.8 59.7
Void shape refer to Figure 2.3b & Figure 2.4 MC2V3 23111.1 90.8
MC3V1 446.48 29.6 MC3V2 3403.4 59.7
MC – Elastic analysis and voids at spring line
L – N – V1 – V2 – V3
Void shape refer to Figure 2.3c & Figure 2.4 MC3V3 10814.4 90.8
Table A.2 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of MC1
Crown Section (MC1)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.16 1.88 661.2 No Void Smooth Condition 2.71 2.30 824.5 Bonded Condition 2.39 2.11 734.4 Remove
Void1 Smooth Condition 2.94 2.50 894.9 Bonded Condition 2.95 2.88 940.9 Remove
Void2 Smooth Condition 3.48 3.22 1085.8 Bonded Condition 3.99 4.18 1320.2 Remove
Void3 Smooth Condition 4.45 4.41 1429.5
Table A.3 Results of Increasing Ratio Based on Table A.2 Ratio Bonded Condition Smooth Condition
Crown Section (MC1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
10.6 12.2 11.0 8.5 8.7 8.5
Remove Void2
36.6 53.2 42.3 28.4 40.0 31.7
Remove Void3
106.0 122.3 99.7 64.3 91.7 73.4
130
Table A.4 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of MC1
Invert Section (MC1)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.22 1.91 677.6 No Void Smooth Condition 2.79 2.28 842.8 Bonded Condition 2.46 2.16 754.8 Remove
Void1 Smooth Condition 2.99 2.55 910.7 Bonded Condition 3.02 2.94 962.1 Remove
Void2 Smooth Condition 3.58 3.22 1106.3 Bonded Condition 4.07 4.17 1323.0 Remove
Void3 Smooth Condition 4.48 4.45 1440.8
Table A.5 Results of Increasing Ratio Based on Table A.4
Ratio Bonded Condition Smooth Condition Invert
Section (MC1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
10.8 13.1 11.4 7.2 11.8 8.1
Remove Void2
36.0 53.9 41.9 28.3 41.2 31.2
Remove Void3
83.3 118.3 96.2 60.6 95.2 70.9
Table A.6 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline of MC1
Springline Section (MC1)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.08 1.26 1069.6 No Void Smooth Condition 3.51 1.80 1123.2 Bonded Condition 3.65 1.75 1194.8 Remove
Void1 Smooth Condition 4.02 2.20 1261.2 Bonded Condition 4.67 2.64 1452.2 Remove
Void2 Smooth Condition 4.92 2.89 1515.4 Bonded Condition 5.74 3.53 1750.6 Remove
Void3 Smooth Condition 5.84 3.67 1773.9
131
Table A.7 Results of Increasing Ratio Based on Table A.6
Ratio Bonded Condition Smooth Condition Springline
Section (MC1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
18.5 38.90 11.8 14.5 22.2 12.3
Remove Void2
51.6 109.5 35.7 40.2 60.6 34.9
Remove Void3
86.4 180.2 63.6 66.4 103.9 58.0
Table A.8 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of MC2
Crown Section (MC2)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.16 1.88 661.2 No Void Smooth Condition 2.71 2.30 824.5 Bonded Condition 2.35 2.07 721.7 Remove
Void1 Smooth Condition 2.82 2.41 859.3 Bonded Condition 2.87 2.70 901.1 Remove
Void2 Smooth Condition 3.25 2.95 1194.5 Bonded Condition 3.73 3.85 1223.8 Remove
Void3 Smooth Condition 4.05 4.00 1298.9
Table A.9 Results of Increasing Ratio Based on Table A.8 Ratio Bonded Condition Smooth Condition
Crown Section (MC2)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
8.8 10.1 9.2 4.0 4.8 4.2
Remove Void2
32.9 43.6 36.3 19.9 28.3 44.9
Remove Void3
72.7 104.8 85.1 49.5 73.9 57.5
132
Table A.10 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of MC2
Invert Section (MC2)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.22 1.91 677.6 No Void Smooth Condition 2.77 2.35 842.7 Bonded Condition 2.41 2.11 738.9 Remove
Void1 Smooth Condition 2.88 2.47 878.3 Bonded Condition 2.94 2.76 922.4 Remove
Void2 Smooth Condition 3.30 3.00 1023.3 Bonded Condition 3.77 3.89 1236.7 Remove
Void3 Smooth Condition 4.09 4.04 1311.8
Table A.11 Results of Increasing Ratio Based on Table A.10
Ratio Bonded Condition Smooth Condition Invert
Section (MC2)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
8.6 10.5 9.0 4.0 5.1 4.2
Remove Void2
32.4 44.5 36.0 19.1 27.7 21.4
Remove Void3
69.8 103.7 82.5 47.7 71.9 55.6
Table A.12 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline of MC2
Springline Section (MC2)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.07 1.26 1064.8 No Void Smooth Condition 3.51 1.80 1123.2 Bonded Condition 3.59 1.73 1173.0 Remove
Void1 Smooth Condition 3.84 2.05 1213.5 Bonded Condition 4.45 2.50 1386.1 Remove
Void2 Smooth Condition 4.54 2.68 1396.7 Bonded Condition 5.30 3.27 1615.4 Remove
Void3 Smooth Condition 5.35 3.35 1626.1
133
Table A.13 Results of Increasing Ratio Based on Table A.12 Ratio Bonded Condition Smooth Condition
Springline Section (MC2)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
16.9 37.3 10.1 9.4 13.9 8.1
Remove Void2
45.0 98.4 30.1 29.3 48.9 24.4
Remove Void3
72.6 159.5 51.6 52.4 86.1 44.8
Table A.14 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @
Crown of MC3
Crown Section (MC3)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.16 1.88 661.2 No Void Smooth Condition 2.71 2.30 824.5 Bonded Condition 2.34 2.07 719.5 Remove
Void1 Smooth Condition 2.81 2.41 857.0 Bonded Condition 2.82 2.65 885.1 Remove
Void2 Smooth Condition 3.13 2.87 973.5 Bonded Condition 3.56 3.60 1155.3 Remove
Void3 Smooth Condition 3.74 3.75 1208.4
Table A.15 Results of Increasing Ratio Based on Table A.14 Ratio Bonded Condition Smooth Condition
Crown Section (MC3)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
8.3 10.1 8.8 3.7 4.8 4.0
Remove Void2
30.6 41.0 33.9 15.5 24.8 18.1
Remove Void3
64.8 91.5 74.7 38.0 63.0 46.6
134
Table A.16 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of MC3
Invert Section (MC3)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.22 1.91 677.6 No Void Smooth Condition 2.77 2.35 842.7 Bonded Condition 2.41 2.11 738.9 Remove
Void1 Smooth Condition 2.87 2.46 875.2 Bonded Condition 2.88 2.71 904.4 Remove
Void2 Smooth Condition 3.19 2.93 1185.9 Bonded Condition 3.60 3.64 1168.2 Remove
Void3 Smooth Condition 3.84 3.75 1225.0
Table A.17 Results of Increasing Ratio Based on Table A.16
Ratio Bonded Condition Smooth Condition Invert
Section (MC3)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
8.6 10.5 9.0 3.6 4.7 3.8
Remove Void2
29.7 41.9 33.3 15.2 24.7 40.7
Remove Void3
62.2 90.6 72.3 38.6 59.6 45.3
Table A.18 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Springline of MC3
Springline Section (MC3)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.07 1.26 1064.8 No Void Smooth Condition 3.51 1.80 1123.2 Bonded Condition 3.58 1.74 1166.3 Remove
Void1 Smooth Condition 3.77 2.08 1180.1 Bonded Condition 4.35 2.44 1355.5 Remove
Void2 Smooth Condition 4.40 2.45 1373.8 Bonded Condition 4.96 3.09 1509.0 Remove
Void3 Smooth Condition 4.97 3.13 1509.1
135
Table A.19 Results of Increasing Ratio Based on Table A.18 Ratio Bonded Condition Smooth Condition
Springline Section (MC3)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
16.6 38.1 9.5 7.4 15.6 5.1
Remove Void2
41.7 93.7 27.3 25.4 36.1 22.4
Remove Void3
61.6 145.2 41.7 41.6 73.9 41.8
Table A.20 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (29.6°) – Bonded Interface Contact Angle (29.6°)
Bonded Interface
MC3V1 MC2V1 MC1V1
Areas ( ) 2mm446.5 1272.7 4141.9
Ratio (%) 0 185.4 828.7 Compressive Stress @ Crown
(%) 0 0.4 2.0
Tensile Stress @ Crown (%)
0 0.5 2.4
Compressive Stress @ Invert (%)
0 0.4 2.5
Tensile Stress @ Invert (%)
0 0.5 2.9
Compressive Stress @ Spring line (%)
0 0.3 2.0
Tensile Stress @ Spring line (%)
0 0.6 1.2
136
Table A.21 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing Areas @ Contact Angle (29.6°) – Full-Slip Interface
Contact Angle (29.6°) Full-Slip Pipe-Soil Interface
MC3V1 MC2V1 MC1V1
Areas ( ) 2mm446.5 1272.7 4141.9
Ratio (%) 0 185.4 828.7 Compressive Stress @ Crown
(%) 0 0.4 4.6
Tensile Stress @ Crown (%)
0 0.4 4.7
Compressive Stress @ Invert (%)
0 0.4 4.2
Tensile Stress @ Invert (%) 0 0.4 3.7 Compressive Stress @ Spring line
(%) 0 1.9 6.6
Tensile Stress @ Spring line (%)
0 1.4 7.3
Table A.22 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (59.7°) – Bonded Interface Contact Angle (59.7°)
Bonded Interface
MC3V2 MC2V2 MC1V2
Areas ( ) 2mm3403.4 8221.8 17435.6
Ratio (%) 0 141.6 412.4 Compressive Stress @ Crown
(%) 0 1.8 4.6
Tensile Stress @ Crown (%)
0 1.9 8.7
Compressive Stress @ Invert (%)
0 2.1 4.9
Tensile Stress @ Invert (%)
0 1.8 8.5
Compressive Stress @ Spring line (%)
0 2.3 7.4
Tensile Stress @ Spring line (%)
0 2.5 8.2
137
Table A.23 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (59.7°) – Full-Slip Interface Contact Angle (59.7°)
Full-Slip Pipe-Soil Interface
MC3V2 MC2V2 MC1V2
Areas ( ) 2mm3403.4 8221.8 17435.6
Ratio (%) 0 141.6 412.4 Compressive Stress @ Crown
(%) 0 3.8 11.2
Tensile Stress @ Crown (%)
0 2.8 12.2
Compressive Stress @ Invert (%)
0 3.5 12.2
Tensile Stress @ Invert (%)
0 2.4 9.9
Compressive Stress @ Spring line (%)
0 3.2 11.8
Tensile Stress @ Spring line (%)
0 9.4 18.0
Table A.24 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (90.8°) – Bonded Interface Contact Angle (90.8°)
Bonded Interface MC3V3 MC2V3 MC1V3
Areas ( ) 2mm10814.4 23111.1 41225.2
Ratio (%) 0 113.7 281.2 Compressive Stress @ Crown
(%) 0 4.8 12.1
Tensile Stress @ Crown (%) 0 7.0 16.1 Compressive Stress @ Invert
(%) 0 4.7 13.1
Tensile Stress @ Invert (%)
0 6.9 14.6
Compressive Stress @ Spring line (%)
0 6.9 17.7
Tensile Stress @ Spring line (%)
0 5.8 14.2
138
Table A.25 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing Areas @ Contact Angle (90.8°) – Full-Slip Interface
Contact Angle (90.8°) Full-Slip Pipe-Soil Interface
MC3V3 MC2V3 MC1V3
Areas ( ) 2mm10814.4 23111.1 41225.2
Ratio (%) 0 113.7 281.2 Compressive Stress @ Crown
(%) 0 8.3 19.0
Tensile Stress @ Crown (%)
0 6.7 17.6
Compressive Stress @ Invert (%)
0 6.5 16.7
Tensile Stress @ Invert (%)
0 7.7 18.7
Compressive Stress @ Spring line (%)
0 7.6 17.5
Tensile Stress @ Spring line (%)
0 7.0 17.3
139
APPENDIX B
FINITE ELEMENT ANALYSIS OF BURIED PIPE TEST – ELASTO-PLASTIC ANALYSIS (VOIDS AT SPRINGLINE)
B.1 DESCRIPTION OF THE FINITE ELEMENT MODELING
The same finite element program, AFENA, was used to study the earth load effect on the
behavior of buried RC rigid pipe with erosion voids in the vicinity. In order to compare
the results between elastic models and elastic-plastic models, the pipe model are same as
described in Appendix A, only assume different soil properties. We set up 3 different
models – MD1, MD2 and MD3 – with different voids information corresponding to MC1,
MC2 and MC3 in Appendix A.
B.2 MODELING OF SOIL INFORMATION
The soil used in all tests was clay with young’s modulus E = 2 MPa, passion ratio r =
0.333 and friction angle θ = . The models were built up gradually by placing the soil
in layers up until full loading. In order to make sure the finite element analysis are
similar to the reality conditions, cohesion of the soil was set to 0.002 kPa and shear
failures would happen during the test.
°20
B.3 MODELING OF EROSION VOIDS INFORMATION
140
Model MD1 has different voids which are MD1V1, MD1V2 and MD1V31. Model MD2
has different voids which are MD2V1, MD2V2 and MD2V3 and Model MD3 has
different voids which are MD3V1, MD3V2 and MD3V3. In order to find the changes of
bending moment and stresses of the RC pipe with different void, we assume that MD1V1,
MD2V1 and MD3V1 have the same contact angle from the center of the rigid pipe
1α = . Same as MD1V2, MD2V2 and MD3V2 have the same contact angle from the
center of the rigid pipe
°6.29
2α = and MD1V3, MD2V3 and MD3V3 have the contact
angle
°7.59
3α = . °8.90
B.4 RESULTS
The test sequences are the same as Appendix A. From theoretical analysis above, the
maximum compressive stress, maximum tensile stresses and maximum bending
moment will occurred at the critical sections of the RC pipe, which are at Crown, Invert
and Springlines.
Model MD1
The results of xxσ at Crown of the RC pipe are shown on Figure 2.11a (bonded
conditions) and 2.11b (smooth condition). The results of xxσ at invert of the RC pipe
are shown on Figure 2.12a (bonded conditions) and 2.12b (smooth condition). The
results of yyσ at springline of the RC pipe are shown on Figure 2.13a (bonded
1 MD1, MD2 and MD3 correspond to Void Set A, Void Set B and Void Set C in Chapter Two.
141
conditions) and 2.13b (smooth condition). The results of xxσ distribution of soil are
shown from Figure B.1 to Figure B.4 and the results of yyσ distribution of soil are
shown from Figure B.5 to Figure B.8.
The results of xxσ at Crown of the RC pipe are shown on Table B.1 and the ratio
changes are shown on Table B.2. The results of xxσ at invert of the RC pipe are shown
on Table B.3 and the ratio changes are shown on Table B.4. The results of yyσ at
springline of the RC pipe are shown on Table B.5 and the ratio changes are shown on
Table B.6.
Model MD2
The results of xxσ at Crown of the RC pipe are shown on Table B.7 and the ratio
changes are shown on Table B.8. The results of xxσ at invert of the RC pipe are shown
on Table B.9 and the ratio changes are shown on Table B.10. The results of yyσ at
springline of the RC pipe are shown on Table B.11 and the ratio changes are shown on
Table B.12.
Model MD3
The results of xxσ at Crown of the RC pipe are shown on Table B.13 and the ratio
changes are shown on Table B.14. The results of xxσ at invert of the RC pipe are shown
on Table B.15 and the ratio changes are shown on Table B.16. The results of yyσ at
springline of the RC pipe are shown on Table B.17 and the ratio changes are shown on
142
Table B.18.
B.5 RESULTS ANALYSIS
The results shown from Table B.1 to B.18 shows that when the pipe and soil are perfectly
adhered together (boned case), the calculated increasing rate of maximum compressive
stress, maximum tensile stresses and maximum bending moment at critical positions are
higher than the full slip case. Increasing rate of maximum compressive stress, maximum
tensile stresses and maximum bending moment are similar at crown and invert positions.
At springline has a higher increasing rate compared to crown and invert positions. The
comparison of results is shown on Figure 2.14. Generally, the increasing rates of plastic
analysis are higher than elastic analysis
The results of comparison of compressive stress and tensile stress of RC pipe at crown of
different models (MD1, MD2 and MD3) are shown on Figure 2.14. For contact angle
90.8°, the compressive stress of RC pipe @ crown of MD1 (bonded interface) increased
142.7%, and the compressive stress of RC pipe @ crown of MD1 (full-slip interface)
increased 212.3%. For contact angle 90.8°, the tensile stress of RC pipe @ crown of
MD1 (bonded interface) increased 214.7%, and the tensile stress of RC pipe @ crown of
MD1 (full-slip interface) increased 155.7%. Therefore, erosion voids would affect more
for bonded interface than full-slip pipe-soil interface. The Figure 2.14 proves that the
contact angle is critical parameter related to the void information, which increase the
stress and moments of the pipes. When the contact angle and void areas increase to
certain level, the compressive stress and tensile stress of RC pipe will stop increasing.
143
144
Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing
Areas at certain angles for different pipe-soil interfaces are shown from Table B.19 to
B.24. The biggest increasing ratios of each table are marked by grey background. For the
worst case, at contact angle 90.8°, when the void area increased 282.22%, the TENSILE
stress of RC pipe @ crown of MD1 (full-slip interface) increased 34.8%. Thus
elastic-plastic finite element analysis prove the same conclusion as the elastic finite
element analysis, that the void area is not very important compared to the contact angle
when considering the design of buried pipe system.
a. Bonded interface b. Smooth interface
Figure B.1 Stress xxσ of Soil – MD1 (No Void)
145
a. Bonded interface b. Smooth interface Figure B.2 Stress xxσ of Soil – MD1 (Remove Void1)
146
a. Bonded interface b. Smooth interface Figure B.3 Stress xxσ of Soil – MD1 (Remove Void2)
147
a. Bonded interface b. Smooth interface Figure B.4 Stress xxσ of Soil – MD1 (Remove Void3)
148
a. Bonded interface b. Smooth interface Figure B.5 Stress yyσ of Soil – MD1 (No Void)
149
a. Bonded interface b. Smooth interface Figure B.6 Stress yyσ of Soil – MD1 (Remove Void1)
150
a. Bonded interface b. Smooth interface
Figure B.7 Stress yyσ of Soil – MD1 (Remove Void2)
151
a. Bonded interface b. Smooth interface
Figure B.8 Stress yyσ of Soil – MD1 (Remove Void3)
152
Table B.1 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of MD1
Table B.2 Results of Increasing Ratio Based on Table B.1
Crown Section (MD1)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.1 1.8 661.2 No Void Smooth Condition 2.4 2.0 824.5 Bonded Condition 2.3 2.0 734.4 Remove
Void1 Smooth Condition 2.7 2.3 894.9 Bonded Condition 3.5 3.4 941.0 Remove
Void2 Smooth Condition 3.8 3.7 1085.8 Bonded Condition 5.2 5.6 1320.2 Remove
Void3 Smooth Condition 5.0 5.1 1429.5
Ratio Bonded Condition Smooth Condition Crown Section (MD1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
9.9 14.7 11.1 14.7 15.9 8.5
Remove Void2
65.7 94.4 42.3 94.4 85.1 31.7
Remove Void3
142.7 214.7 99.7 215.3 155.7 73.4
Table B.3 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert
of MD1 Invert Section
(MD1) )(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.2 1.8 677.6 No Void Smooth Condition 2.5 2.1 842.8 Bonded Condition 2.4 2.1 754.8 Remove
Void1 Smooth Condition 2.8 2.4 910.7 Bonded Condition 3.5 3.4 962.1 Remove
Void2 Smooth Condition 3.9 3.7 1106.3 Bonded Condition 5.3 5.7 1330.0 Remove
Void3 Smooth Condition 5.1 5.3 1440.8
153
Table B.4 Results of Increasing Ratio Based on Table B.3 Ratio Bonded Condition Smooth Condition Invert
Section (MD1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
9.2 11.3 11.4 14.4 14.6 8.1
Remove Void2
61.0 57.5 41.9 89.0 78.6 31.2
Remove Void3
142.7 104.9 96.2 213.3 155.3 71.0
Table B.5 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Spring
line of MD1 Springline Section
(MD1) )(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.0 1.3 1069.6 No Void Smooth Condition 3.2 1.7 1123.2 Bonded Condition 3.6 1.8 1194.8 Remove
Void1 Smooth Condition 3.9 2.2 1261.2 Bonded Condition 5.2 3.0 1452.2 Remove
Void2 Smooth Condition 5.1 3.1 1515.4 Bonded Condition 6.5 4.1 1750.6 Remove
Void3 Smooth Condition 5.7 3.7 1773.9
Table B.6 Results of Increasing Ratio Based on Table B.5
Ratio Bonded Condition Smooth Condition Springline
Section (MD1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
21.7 38.6 11.8 23.6 28.6 12.3
Remove Void2
74.6 137.0 35.7 61.6 83.3 34.9
Remove Void3
117.1 221.3 63.6 79.6 117.9 58.0
154
Table B.7 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of MD2
Table B.8 Results of Increasing Ratio Based on Table B.7
Crown Section (MD2)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.1 1.8 661.2 No Void Smooth Condition 2.4 2.0 824.5 Bonded Condition 2.2 2.0 721.7 Remove
Void1 Smooth Condition 2.6 2.1 859.3 Bonded Condition 3.1 3.0 901.1 Remove
Void2 Smooth Condition 3.3 3.0 1194.6 Bonded Condition 4.6 4.9 1223.8 Remove
Void3 Smooth Condition 4.8 5.0 1298.9
Ratio Bonded Condition Smooth Condition Crown Section (MD2)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
5.2 10.7 9.2 7.0 6.5 4.2
Remove Void2
47.0 67.8 36.3 35.1 47.3 44.9
Remove Void3
117.4 176.3 85.1 98.0 146.3 57.5
Table B.9 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of MD2
Invert Section (MD2)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.2 1.8 677.6 No Void Smooth Condition 2.5 2.1 842.7 Bonded Condition 2.3 2.0 738.9 Remove
Void1 Smooth Condition 2.6 2.2 878.3 Bonded Condition 3.1 3.0 922.4 Remove
Void2 Smooth Condition 3.3 3.0 1023.3 Bonded Condition 4.7 4.9 1236.7 Remove
Void3 Smooth Condition 4.8 5.0 1311.8
155
Table B.10 Results of Increasing Ratio Based on Table B.9
Ratio Bonded Condition Smooth Condition Invert
Section (MD2)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
5.1 10.5 9.0 6.5 5.3 4.2
Remove Void2
43.6 64.1 36.0 32.8 43.7 21.4
Remove Void3
115.6 171.8 82.5 95.1 141.6 55.6
Table B.11 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Spring line of MD2
Springline Section (MD2)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.0 1.3 1064.8 No Void Smooth Condition 3.2 1.7 1123.2 Bonded Condition 3.4 1.7 1173.0 Remove
Void1 Smooth Condition 3.6 2.0 1213.5 Bonded Condition 4.8 2.7 1386.1 Remove
Void2 Smooth Condition 4.7 2.7 1396.7 Bonded Condition 5.9 3.8 1615.4 Remove
Void3 Smooth Condition 5.7 3.7 1626.1
Table B.12 Results of Increasing Ratio Based on Table B.11 Ratio Bonded Condition Smooth Condition
Springline Section (MD2)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
13.4 35.4 10.1 14.2 18.5 8.1
Remove Void2
60.5 115.0 30.1 48.4 63.1 24.4
Remove Void3
98.0 196.9 51.6 80.5 118.5 44.8
156
Table B.13 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of MD3
Table B.14 Results of Increasing Ratio Based on Table B.12
Crown Section (MD3)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.1 1.8 661.2 No Void Smooth Condition 2.4 2.0 824.5 Bonded Condition 2.2 1.9 719.5 Remove
Void1 Smooth Condition 2.6 2.1 857.0 Bonded Condition 2.9 2.8 885.1 Remove
Void2 Smooth Condition 3.1 2.8 973.5 Bonded Condition 4.3 4.4 1155.3 Remove
Void3 Smooth Condition 4.4 4.5 1208.4
Ratio Bonded Condition Smooth Condition Crown Section (MD3)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
4.7 9.6 8.8 6.6 6.0 4.0
Remove Void2
38.0 57.6 33.9 26.5 37.3 18.1
Remove Void3
101.4 150.3 74.7 81.8 121.9 46.6
Table B.15 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of MD2
Invert Section (MD3)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.2 1.8 677.6 No Void Smooth Condition 2.5 2.1 842.7 Bonded Condition 2.3 2.0 738.9 Remove
Void1 Smooth Condition 2.6 2.2 875.2 Bonded Condition 2.9 2.8 904.4 Remove
Void2 Smooth Condition 3.1 2.8 1185.9 Bonded Condition 4.3 4.4 1168.2 Remove
Void3 Smooth Condition 4.4 4.4 1225.0
157
Table B.16 Results of Increasing Ratio Based on Table B.15
Ratio Bonded Condition Smooth Condition Invert
Section (MD3)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
5.0 10.5 9.0 6.1 5.3 3.8
Remove Void2
34.9 55.3 33.3 24.7 35.4 40.7
Remove Void3
97.3 143.7 72.3 76.1 114.1 45.3
Table B.17 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Spring line of MD3
Springline Section (MD3)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.0 1.3 1064.8 No Void Smooth Condition 3.2 1.7 1123.2 Bonded Condition 3.4 1.7 1166.3 Remove
Void1 Smooth Condition 3.6 2.0 1180.1 Bonded Condition 4.6 2.6 1355.5 Remove
Void2 Smooth Condition 4.5 2.6 1373.8 Bonded Condition 5.5 3.5 1509.0 Remove
Void3 Smooth Condition 5.5 3.5 1509.1
Table B.18 Results of Increasing Ratio Based on Table B.16 Ratio Bonded Condition Smooth Condition
Springline Section (MD3)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
13.7 31.5 9.5 13.5 17.3 5.1
Remove Void2
52.8 105.5 27.3 40.9 56.0 22.4
Remove Void3
85.3 172.4 41.7 74.2 106.6 41.8
158
Table B.19 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing Areas @ Contact Angle (29.6°) – Bonded Interface
Contact Angle (29.6°) Bonded Interface
MD3V1 MD2V1 MD1V1
Areas ( ) 2mm446.5 1272.7 4141.9
Ratio (%) 0 185.4 828.7 Compressive Stress @ Crown
(%) 0 0.5 4.9
Tensile Stress @ Crown (%)
0 1.0 4.6
Compressive Stress @ Invert (%)
0 0.4 3.9
Tensile Stress @ Invert (%)
0 0.5 3.5
Compressive Stress @ Spring line (%)
0 0.3 7.1
Tensile Stress @ Spring line (%)
0 3.0 5.4
Table B.20 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (29.6°) – Full-Slip Interface Contact Angle (29.6°)
Full-Slip Pipe-Soil Interface
MD3V1 MD2V1 MD1V1
Areas ( ) 2mm446.5 1272.7 4141.9
Ratio (%) 0 185.4 828.7 Compressive Stress @ Crown
(%) 0 0.4 5.4
Tensile Stress @ Crown (%) 0 0.5 9.4 Compressive Stress @ Invert
(%) 0 0.4 5.0
Tensile Stress @ Invert (%)
0 0.5 8.8
Compressive Stress @ Spring line (%)
0 0.6 8.9
Tensile Stress @ Spring line (%)
0 1.0 9.6
159
Table B.21 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing Areas @ Contact Angle (59.7°) – Bonded Interface
Contact Angle (59.7°) Bonded Interface
MD3V2 MD2V2 MD1V2
Areas ( ) 2mm3403.4 8221.8 17435.6
Ratio (%) 0 141.6 412.4 Compressive Stress @ Crown
(%) 0 5.0 14.2
Tensile Stress @ Crown (%)
0 6.5 23.3
Compressive Stress @ Invert (%)
0 6.5 19.4
Tensile Stress @ Invert (%)
0 5.7 21.7
Compressive Stress @ Spring line (%)
0 7.1 14.2
Tensile Stress @ Spring line (%)
0 4.6 15.3
Table B.22 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (59.7°) – Full-Slip Interface Contact Angle (59.7°)
Full-Slip Pipe-Soil Interface
MD3V2 MD2V2 MD1V2
Areas ( ) 2mm3403.4 8221.8 17435.6
Ratio (%) 0 141.6 412.4 Compressive Stress @ Crown
(%) 0 6.9 25.2
Tensile Stress @ Crown (%) 0 7.3 34.8 Compressive Stress @ Invert
(%) 0 6.5 26.3
Tensile Stress @ Invert (%)
0 6.1 31.9
Compressive Stress @ Spring line (%)
0 5.4 14.8
Tensile Stress @ Spring line (%)
0 4.6 17.6
160
Table B.23 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe with Increasing Areas @ Contact Angle (90.8°) – Bonded Interface
Contact Angle (90.8°) Bonded Interface
MD3V3 MD2V3 MD1V3
Areas ( ) 2mm10814.4 23111.1 41225.2
Ratio (%) 0 113.7 281.2 Compressive Stress @ Crown
(%) 0 6.9 17.1
Tensile Stress @ Crown (%) 0 10.4 26.0 Compressive Stress @ Invert
(%) 0 9.3 23.0
Tensile Stress @ Invert (%)
0 11.6 28.6
Compressive Stress @ Spring line (%)
0 6.9 17.3
Tensile Stress @ Spring line (%)
0 9.0 17.9
Table B.24 Increasing Ratio of Compressive Stress and Tensile Stress of RC Pipe
with Increasing Areas @ Contact Angle (90.8°) – Full-Slip Interface Contact Angle (90.8°)
Full-Slip Pipe-Soil Interface
MD3V3 MD2V3 MD1V3
Areas ( ) 2mm10814.4 23111.1 41225.2
Ratio (%) 0 113.7 281.2 Compressive Stress @ Crown
(%) 0 6.9 25.2
Tensile Stress @ Crown (%)
0 7.3 34.8
Compressive Stress @ Invert (%)
0 6.5 26.3
Tensile Stress @ Invert (%)
0 6.1 31.9
Compressive Stress @ Spring line (%)
0 5.4 14.7
Tensile Stress @ Spring line (%)
0 4.6 17.6
161
APPENDIX C
FINITE ELEMENT ANALYSIS OF BURIED RC PIPE TEST – ELASTO-PLASTIC ANALYSIS (VOIDS AT INVERT)
C.1 DESCRIPTION OF THE FINITE ELEMENT MODELING
The same finite element program, AFENA, was used to study the earth load effect on the
behavior of buried RC rigid pipe with erosion voids in the vicinity. To calculate the pipe
model, two-dimensional plane strain analysis was carried out. We set up 1 model – ME1
– with different voids information under invert. The meshes used in this numerical
analysis are shown on Figure 2.15. Only half the biaxial test cell was analyzed due to
symmetry of the problem. The side wall interface between the soil and the test cell was
modeled as smooth and rigid. Close to eleven hundred six-node triangles were used to
model the backfill soil, and finer elements were used in the vicinity of the pipe.
The modulus of the RC pipe has no effect on the calculated response, since it is
essentially rigid compared to soil. Joint elements, 0.25mm in length, were placed between
the pipe and the soil to model the interface. It has negligible effect on the results. For the
bonded pipe-soil condition, these joint elements are prescribed with both normal and
shear stiffness. Conversely, high normal stiffness but negligible shear stiffness was
specified for the full-slip condition. The stiffness value of the joints with high stiffness
was chosen so that the joints are stiff enough so that slipping does not occur. Additional
analyses using different stiffness values were performed to ensure numerically stability
162
and convergence of the result. The vertical pressure was applied at the top of the backfill
soil using of 100 kPa.
C.2 MODELING OF SOIL INFORMATION
The soil used in all analyses was clay with the young’s modulus E = 2 MPa, passion ratio
r = 0.333 and friction angle θ = . The models were built up gradually by placing the
soil in layers up until full loading. In order to make sure the finite element analysis are
similar to the reality conditions, Cohesion of the soil was set to 0.002 kPa and shear
failures would happen during the test.
°20
C.3 RESULTS
Test sequences are the same as Appendix A. From theories analysis above, the maximum
compressive stress, maximum tensile stresses and maximum bending moment will
occurred at the critical sections of the RC pipe, which are at Crown, Invert and
Springline.
Model C1
The results of xxσ at Crown of the RC pipe are shown on Figure 2.16a (bonded
conditions) and 2.16b (smooth condition). The results of xxσ at invert of the RC pipe
are shown on Figure 2.17a (bonded conditions) and 2.17b (smooth condition). The results
of yyσ at springline of the RC pipe are shown on Figure 2.18a (bonded conditions) and
163
164
xx
From the results shown Table C.1 to C.6, they shows that when the pipe and soil are
perfectly adhered together (boned case), the calculated changing rate of maximum
compressive stress, maximum tensile stresses and maximum bending moment at critical
positions are higher than the full slip case. The maximum compressive stress, maximum
tensile stresses and maximum bending moment at critical positions will change of
opposite sign.
2.18b (smooth condition). The results of σ distribution of soil are shown from Figure
C.1 to Figure C.4, and the results of yyσ distribution of soil are shown from Figure C.5
to Figure C.8. The results of xxσ at Crown of the RC pipe are shown on Table C.1 and
the ratio changes are shown on Table C.2. The results of xxσ at invert of the RC pipe are
shown on Table C.3 and the ratio changes are shown on Table C.4. The results of
C.4 RESULTS ANALYSIS
yyσ at
springline of the RC pipe are shown on Table C.5 and the ratio changes are shown on
Table C.6.
Figure C.1 Stress xxσ of Soil – Bonded Condition (No Void) - Left
Figure C.2 Stress xxσ of Soil – Bonded Condition (Remove Void1) - Right
165
Figure C.3 Stress xxσ of Soil – Bonded Condition Remove Void2) - Left
Figure C.4 Stress xxσ of Soil – Bonded Condition Remove Void3) - Right
166
Figure C.5 Stress yyσ of Soil – Bonded Condition (No Void) - Left
Figure C.6 Stress yyσ of Soil – Bonded Condition (Remove Void1) - Right
167
Figure C.8 Stress yyσ of Soil – Bonded Condition (Remove Void3) - Right
Figure C.7 Stress yyσ of Soil – Bonded Condition (Remove Void2) - Left
168
Table C.1 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Crown of ME1
Crown Section (ME1)
)(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.1 1.8 643.0 No Void Smooth Condition 2.4 2.1 731.6 Bonded Condition 1.5 1.1 455.4 Remove
Void1 Smooth Condition 1.6 1.2 473.5 Bonded Condition 0.7 0.0 426.2 Remove
Void2 Smooth Condition 0.3 -0.3 93.5 Bonded Condition -0.1 -1.0 -556.2 Remove
Void3 Smooth Condition -0.8 -1.5 -471.9
Table C.2 Results of Increasing Ratio Based on Table C.1 Ratio Bonded Condition Smooth Condition
Crown Section (ME1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
-27.6 -37.7 -29.2 -33.9 -42.8 -35.4
Remove Void2
-67.6 -99.5 -33.8 -87.0 -112.0 -87.2
Remove Void3
-104.3 -155.7 -186.5 -133.1 -171.6 -164.5
Table C.3 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @ Invert of
ME1 Invert Section
(ME1) )(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 2.2 1.9 661.2 No Void Smooth Condition 2.5 2.1 754.4 Bonded Condition 0.7 0.1 348.6 Remove
Void1 Smooth Condition 0.5 -0.3 136.1 Bonded Condition -0.6 -1.3 -464.7 Remove
Void2 Smooth Condition -0.8 -1.7 -542.8 Bonded Condition -1.2 -2.0 -594.6 Remove
Void3 Smooth Condition -1.3 -2.1 -640.6
169
Table C.4 Results of Increasing Ratio Based on Table C.3 Ratio Bonded Condition Smooth Condition Invert
Section (ME1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
-69.4 -96.3 -47.2 -81.9 -114.0 -82.0
Remove Void2
-125.5 -171.7 -170.4 -131.3 -179.7 -172.0
Remove Void3
-155.6 -204.3 -190.0 -151.8 -201.5 -185.0
Table C.5 Compressive Stresses, Tensile Stresses and Moment of RC Pipe @
Springline of ME1 Springline Section
(ME1) )(max nCompressioσ
(MPa)
)(min Tensionσ
(Mpa)
Moment
610− MN.m/m
Bonded Condition 3.0 1.3 1025.1 No Void Smooth Condition 3.2 1.7 1010.5 Bonded Condition 2.1 0.7 780.1 Remove
Void1 Smooth Condition 2.0 0.8 691.5 Bonded Condition 0.5 -0.3 152.0 Remove
Void2 Smooth Condition 0.1 -0.4 178.9 Bonded Condition -0.8 -1.1 -335.5 Remove
Void3 Smooth Condition -1.2 -1.3 -405.1
Table C.6 Results of Increasing Ratio Based on Table C.5 Ratio Bonded Condition Smooth Condition
Springline Section (ME1)
maxσ
(%)
minσ
(%)
Moment (%) maxσ
(%)
minσ
(%)
Moment (%)
No Void 0 0 0 0 0 0 Remove Void1
-30.7 -45.3 -23.9 -38.6 -54.4 -31.5
Remove Void2
-83.7 -121.9 -85.2 -96.2 -121.9 -82.3
Remove Void3
-127.0 -187.5 -132.7 -136.4 -187.5 -140.1
170
APPENDIX D
SYMBOLS AND ACRONYMS
pA = cross-section area per unit length of pipe BF = bedding factor; empirical moment correction factor based on burial condition c = Distance from the neutral axis of the liner to the outside liner surface C = hoop compression stiffness of the soil relative to the pipe, Hoeg (1968)
pE = Young’s Modulus of pipe
sE = Young’s Modulus of soil F = flexural stiffness of the soil relative to the pipe, Hoeg (1968) g = acceleration due to gravity
pI = second moment of area of pipe wall per unit length of pipe K = coefficient of lateral earth pressure
crM = bending moment per unit length at the crown of pipe
spM = bending moment per unit length at the springline of pipe
crN = thrust per unit length at the crown of pipe
spN = thrust per unit length at the springline of pipe OD = outer diameter of pipe PS = pipe stiffness PVC = poly(vinyl chloride) r = Radius of pipe
linerR = average radius of the liner SF = safety factor t = thickness of the pipe wall
crW = vertical parallel plate load applied to concrete pipe which includes limiting crack
vW = vertical force transmitted to the pipe by the overlying soil γ = unit weight of soil
crε = tensile bending strain on the outside of the liner at the crown
inε = tensile bending strain on the outside of the liner at the springline
hD∆ = change in horizontal pipe diameter
vD∆ = change in vertical pipe diameter φ = angle of internal friction of the soil
pν = Poisson’s ratio of the pipe
171
sν = Poisson’s ratio of the backfill soil σ = radial earth pressures acting on the pipe
hσ = horizontal earth pressures
vσ = vertical earth pressures
xxσ = earth pressures at x direction
yyσ = earth pressures at y direction
0σ = uniform component of radial earth pressures acting on the pipe
2σ = non-uniform component of radial earth pressures acting on the pipe τ = shear stress acting on the pipe
2τ = non-uniform component of shear stress acting on the pipe
172